Journal of Lie Theory Volume 13 (2003) 91–132 \begin{matrix}c \\ ◯ \end{matrix} 2003 Heldermann Verlag

Lie Algebras Graded by the Root System BC_{1}

Georgia Benkart^{∗} and Oleg Smirnov ^{∗∗}

Communicated by K.-H. Neeb

Abstract. We classify the Lie algebras that are graded by the nonreduced root system BC_{1} and determine their central extensions, derivations, and invariant forms.

1. Introduction

Lie algebras graded by reduced root systems

1.1 Let 𝔤 be a finite-dimensional split simple Lie algebra over a field F of characteristic zero with root space decomposition 𝔤 = 𝔥 ⊕ ⨁_{μ∈Δ} 𝔤_{μ} relative to a split Cartan subalgebra 𝔥. Such a Lie algebra 𝔤 is the F-analogue of a finitedimensional simple complex Lie algebra. Motivated by the study of intersection matrix Lie algebras which arose in Slodowy’s work [33] on singularities, Berman and Moody [19] initiated the investigation of Lie algebras graded by the root system Δ. Following them we say

Definition 1.2. A Lie algebra L over F is graded by the (reduced) root system Δ or is Δ-graded if

(i) L contains as a subalgebra a finite-dimensional split simple Lie algebra 𝔤 = 𝔥 ⊕ ⨁_{μ∈Δ} 𝔤_{μ} whose root system is Δ relative to a split Cartan subalgebra 𝔥 = 𝔤_{0} ;

(ii) L = ⨁_{μ∈Δ∪{0}} L_{μ}, where L_{μ} = {x ∈ L | [h, x] = μ(h)x for all h ∈ 𝔥}

for μ ∈ Δ ∪ {0}; and (iii) L_{0} = ∑_{μ∈Δ}[L_{μ}, L_{−μ}].

The subalgebra 𝔤 is called the grading subalgebra of L.

ISSN 0949–5932 / $2.50 \begin{matrix}c \\ ◯ \end{matrix} Heldermann Verlag 1.3 The Δ-graded Lie algebras have been determined up to central extensions by Berman and Moody [19] for Δ = A_{r}, (r ≥ 2), D_{r}, E_{6} , E_{7} , and E_{8} ; by Benkart and Zelmanov ([36], [14], [15]) for Δ = A_{1} , B_{r}, C_{r}, F_{4} , and G_{2} (see also [7] for C_{2} ); and by Neher [29], who studied the Δ-graded Lie algebras for Δ ≠ E_{8} , F_{4} , G_{2} by Jordan algebra methods. Their central extensions have been described by Allison, Benkart, Gao [7] (see also [16], [17], and [18] for earlier work on types A_{r}, D_{r}, E_{6} , E_{7} , E_{8} ). As a result, the Δ-graded Lie algebras are classified completely up to isomorphism. The derivations and invariant forms of all Δ-graded Lie algebras have been computed in [12], and a criterion for their simplicity can be found in [13].

∗ The first author gratefully acknowledges support from National Science Foundation Grants #DMS–9622447 and #DMS–9970119 and from the Ellentuck Fund at the Institute for Advanced Study, Princeton.

∗∗ The second author acknowledges partial support from National Science and Engineering Research Council of Canada.

In this work we address the problem of studying the Lie algebras graded by the nonreduced root systems BC_{r}, and in particular, by the root system BC_{1} .

Lie algebras graded by the nonreduced root systems BC_{r}

1.4 Assume 𝔤 is a split “simple”* Lie algebra whose root system relative to a split Cartan subalgebra 𝔥 is of type B_{r}, C_{r}, or D_{r} for some r ≥ 1. Then 𝔤 = 𝔥 ⊕ ⨁_{μ∈Δ_{X}} 𝔤_{μ} where X = B, C, or D, and

(1.5)  \begin{matrix}Δ_{B} = \{±ϵ_{i} ± ϵ_{j} | 1 ≤ i ≠ j ≤ r\} ∪ \{±ϵ_{i} | i = 1, …, r\}, \\ Δ_{C} = \{±ϵ_{i} ± ϵ_{j} | 1 ≤ i ≠ j ≤ r\} ∪ \{±2ϵ_{i} | i = 1, …, r\}, \\ Δ_{D} = \{±ϵ_{i} ± ϵ_{j} | 1 ≤ i ≠ j ≤ r\}. \end{matrix}

The set

(1.6)  Δ = {±ϵ_{i} ± ϵ_{j} | 1 ≤ i ≠ j ≤ r} ∪ {±ϵ_{i}, ±2ϵ_{i} | i = 1, …, r},

which is contained in the dual space 𝔥^{∗} of 𝔥, is a root system of type BC_{r} in the sense of Bourbaki [20, Chap. VI]. It is nonreduced since both ϵ_{i} and 2ϵ_{i} are roots. The following notion of a Lie algebra graded by the root system BC_{r} was introduced in [8]:

Definition 1.7. A Lie algebra L over a field F of characteristic zero is graded by the root system BC_{r} or is BC_{r} -graded if

(i) L contains as a subalgebra a finite-dimensional split “simple” Lie algebra 𝔤 = 𝔥 ⊕ ⨁_{μ∈Δ_{X}} 𝔤_{μ} whose root system relative to a split Cartan subalgebra 𝔥 = 𝔤_{0} is Δ_{X}, X = B, C, or D as in (1.5);

(ii) L = ⨁_{μ∈Δ∪{0}} L_{μ}, where L_{μ} = {x ∈ L | [h, x] = μ(h)x for all h ∈ 𝔥}

for μ ∈ Δ ∪ {0} and Δ is the root system BC_{r} in (1.6); and (iii) L_{0} = ∑_{μ∈Δ}[L_{μ}, L_{−μ}].

It follows from (ii) that [L_{μ}, L_{ν}] ⊆ L_{μ+ν} if μ + ν ∈ Δ ∪ {0} and [L_{μ}, L_{ν}] = (0) otherwise.

1.8 We refer to 𝔤 as the grading subalgebra of L and say L is BC_{r} -graded with grading subalgebra 𝔤 of type X_{r} (where X = B, C, or D) to mean that the

* In all cases but two, 𝔤 is simple. When 𝔤 is of type D_{2} = A_{1}× A_{1} then 𝔤 ≅ 𝔰𝔩_{2} ⊕ 𝔰𝔩_{2} , and when 𝔤 is of type D_{1} , then 𝔤 = 𝔥, which is 1-dimensional. root system of 𝔤 is of type X_{r}. The classification of finite-dimensional simple Lie algebras over nonalgebraically closed fields of characteristic zero which contain an 𝔰𝔩_{2} subalgebra (or equivalently, an ad-nilpotent element) involves Lie algebras that are BC_{r} -graded relative to a split Cartan subalgebra. For such Lie algebras, it is possible to select a grading subalgebra of any one of the types B_{r}, C_{r}, or D_{r}. In [32], Seligman works with a grading subalgebra of type C_{r}. However, for (possibly infinite-dimensional and possibly non-simple) general BC_{r} -graded Lie algebras, the existence of a grading subalgebra of type B_{r} or D_{r} does not guarantee the existence of a grading subalgebra of type C_{r}. For that reason we have broadened the concept of a BC_{r} -graded Lie algebra here and previously in [8] to include those of types B_{r} and D_{r}.

1.9 When viewed as a 𝔤-module under the adjoint action, a BC_{r} -graded Lie algebra L decomposes into a direct sum of finite-dimensional irreducible 𝔤-modules. There is one possible isotypic component corresponding to each root length and one corresponding to 0 (the sum of the trivial 𝔤-modules). Thus, if r ≥ 2 and 𝔤 is not of type D_{2} , there are up to four isotypic components which can be parametrized by subspaces A, B, C, D, so that the decomposition is given by

(1.10)  L = (𝔤 ⊗ A) ⊕ (𝔰 ⊗ B) ⊕ (V ⊗ C) ⊕ D.

Here V is an n-dimensional space with a nondegenerate bilinear form ( |) which is symmetric of maximal Witt index or is skew-symmetric. We write (u|v) = ζ(v|u) where ζ = 1 if the form is symmetric, and ζ = −1 if it is skew-symmetric. Then 𝔤 can be realized as the space

(1.11)  𝔤 = {x ∈ End V | (xu|v) = −(u|xv) for all u, v ∈ V }

of all skew-symmetric transformations relative to the form. If ζ = −1 (necessarily n = 2r), then 𝔤 is a split simple Lie algebra of type C_{r}. If ζ = 1, then 𝔤 is of type B_{r} when n = 2r + 1, and 𝔤 is of type D_{r} when n = 2r. The 𝔤-module 𝔰 can be identified with the space,

(1.12)  𝔰 = {s ∈ End V | (su|v) = (u|sv) for all u, v ∈ V, and tr(s) = 0},

of all symmetric transformations of trace zero relative to the form. In the exceptional case that L is BC_{2} -graded of type D_{2} , then

(1.13 D_{2} ) L = (𝔤^{(1)} ⊗ A^{(1)}) ⊕ (𝔤^{(2)} ⊗ A^{(2)}) ⊕ (𝔰 ⊗ B) ⊕ (V ⊗ C) ⊕ D,

where 𝔤 = 𝔤^{(1)}⊕𝔤^{(2)} ≅ 𝔰𝔩_{2}⊕𝔰𝔩_{2} , V is a 4-dimensional space with a nondegenerate symmetric bilinear form ( |) of maximal Witt index, and 𝔤, 𝔰 are as above.

1.14 Any Lie algebra having a decomposition as in (1.10) and satisfying (iii) of Definition 1.7 is BC_{r} -graded. In particular, a Lie algebra graded by the root system B_{r}, C_{r}, or D_{r} is also graded by BC_{r}. In the first case, 𝔤 is a split simple Lie algebra of type B_{r} and B = (0); while in the second, 𝔤 is a split simple Lie algebra of type C_{r} and C = (0); and in the third, 𝔤 is a split simple Lie algebra of type D_{r} and B = (0) = C. The twisted affine Lie algebras of type A_{2r}^{(2)} , r ≥ 1, which have a realization as (𝔤⊗F[t^{±2}])⊕(𝔰⊗tF[t^{±2}])⊕Fz, where 𝔤 is of type B_{r}, F[t^{±1}] is the algebra of Laurent polynomials in t, and Fz is a nonsplit center, are BC_{r} -graded of type B_{r} (see [25], Chap. 8). The twisted affine algebras of type A_{2r−1}^{(2)} , r ≥ 2, can be realized as (𝔤 ⊗ F[t^{±2}]) ⊕ (𝔰 ⊗ tF[t^{±2}]) ⊕ Fz, where 𝔤 is of type C_{r} and z is central, so they are C_{r} -graded, hence BC_{r} -graded. Similarly, the twisted affine Lie algebras of type D_{r+1}^{(2)} can be identified with (𝔤 ⊗ F[t^{±2}]) ⊕ (V ⊗ tF[t^{±2}]) ⊕ Fz, where 𝔤 is of type B_{r}, so they are B_{r} -graded and hence, BC_{r} -graded.

The extended affine Lie algebras (previously termed quasisimple Lie algebras) are natural generalizations of the affine Lie algebras. The cores of extended affine Lie algebras of type BC_{r} are Lie algebras which are BC_{r} -graded, (see [6], [24]). Thus, the results of [8] and of our present study on BC_{1} -graded Lie algebras are expected to shed light on the extended affine Lie algebras of type BC_{r} and to facilitate their classification. As we have mentioned earlier, BC_{r} -graded Lie algebras arise naturally in the classification of finite-dimensional simple Lie algebras over nonalgebraically closed fields of characteristic zero (see [32] and in particular, see [1], [5] for BC_{1} -graded algebras). The “odd symplectic” Lie algebras of Gelfand and Zelevinsky [23] and of Proctor [30] and certain of the intersection matrix algebras of Slodowy [33] provide further interesting examples of BC_{r} -graded Lie algebras. Consequently, the notion of a BC_{r} -graded Lie algebra is an important unifying concept encompassing a diverse array of Lie algebras.

1.15 The classification problem for BC_{r} -graded Lie algebras amounts to determining the coordinating spaces A, B, C, D (or A^{(1)}, A^{(2)}, B, C, D) and the multiplication in L. This is accomplished in [8] for r ≥ 2. The BC_{1} case differs from the higher rank cases in that (as we discuss below) the Lie algebras graded by BC_{1} have many additional degeneracies, and because of that, they require more general coordinates. Indeed, the phenomenon that small rank algebras exhibit degeneracies already is very apparent in the work [8], where roughly half the monograph is devoted to the Lie algebras graded by the root system BC_{2} .

1.16 In this paper we classify the BC_{1} -graded Lie algebras and determine their derivations, central extensions, and invariant forms. An important role is played by 5-graded Lie algebras. There are several algebraic structures connected with 5graded Lie algebras - structurable algebras, J-ternary algebras, and Kantor pairs. All are generalizations of Jordan algebras. For each of them there is a Lie algebra construction which generalizes the well-known Tits-Kantor-Koecher construction and which produces 5-graded Lie algebras. The need for more general coordinates for BC_{1} -graded Lie algebras requires us to introduce a new class of algebraic structures, termed Jordan-Kantor pairs, and to give an even more general Tits-Kantor-Koecher construction of an associated Lie algebra. Simple Z-graded Lie algebras have been studied extensively (e.g., [26], [35], [34]). In [35] they are shown to arise from associative algebras or to be finite-dimensional exceptional Lie algebras (possibly after a change of the base field). The 5-graded Lie algebras we encounter in this work are not assumed to be simple or finite-dimensional, and because of this, their description involves the more general algebraic structures defined in (3.1).

1.17 Our main results are

(i) an introduction of the notion of a Jordan-Kantor pair (in (3.1)) and its Tits-Kantor-Koecher Lie algebra (in Section 4), and an investigation of their properties;

(ii) the classification of BC_{1} graded Lie algebras of type D_{1} (Theorem 5.24),

of type C_{1} (Theorem 6.34), and of type B_{1} (Theorem 7.20); (iii) a determination of the derivations of all BC_{1} -graded Lie algebras (The-

orem 8.6); (iv) a description of the invariant forms on all BC_{1} -graded Lie algebras

(Theorem 9.10). 1.18 Some results in this paper have been presented at the International Conference on Jordan Structures, Universidad de Málaga, Spain, June 1997, and the Workshop in Algebra, Ottawa-Carleton Institute of Mathematics and Statistics, Canada, September 1996. We take this opportunity to thank the organizers of those conferences. Also our special thanks go to Bruce Allison for several helpful discussions.

2. BC_{1} -graded Lie algebras.

2.1 Suppose L is a BC_{1} -graded Lie algebra with grading subalgebra 𝔤, and consider the decomposition of L into irreducible 𝔤-modules relative to ad 𝔤. Assume V, 𝔤, 𝔰 are as in Section 1 (see (1.11) and (1.12)). When the grading subalgebra 𝔤 is of type B_{1} , 𝔤 ≅ V as 𝔤-modules. When 𝔤 is of type C_{1} , then 𝔰 = (0); and when 𝔤 is of type D_{1} , 𝔤 = Fh. This means that any BC_{1} -graded Lie algebra has one of the following forms,

(2.2)  L = \{ \begin{matrix}(𝔤 ⊗ A) ⊕ (𝔰 ⊗ B) ⊕ D & type B_{1}, \\ (𝔤 ⊗ A) ⊕ (V ⊗ C) ⊕ D & type C_{1}, \\ L_{−2} ⊕ L_{−1} ⊕ L_{0} ⊕ L_{1} ⊕ L_{2} & type D_{1}. \end{matrix}

For D_{1} , 𝔤 = Fh, and so the decomposition into 𝔤-modules is simply the decomposition into eigenspaces relative to ad h. For all types, we identify the Lie algebra 𝔤 with 𝔤 ⊗ 1 so that [x, y] corresponds to [x ⊗ 1, y ⊗ 1] = [x, y] ⊗ 1.

2.3 When the grading subalgebra is of type B_{1} or C_{1} , then 𝔤 ≅ 𝔰𝔩_{2} , and L decomposes into irreducible 𝔤-modules of dimensions 1, 3, 5 in the B-case and of dimensions 1, 2, 3 in the C-case. In writing (2.2), we have collected the isomorphic 𝔤-summands, so in the B-case, 𝔰 ⊗ B is the sum of all the 5-dimensional 𝔤modules, and in the C-case, V ⊗C is the sum of all the 2-dimensional 𝔤-modules. For both cases, 𝔤 ⊗ A is the sum of all the 3-dimensional 𝔤-modules, and D is the sum of the 1-dimensional 𝔤-modules. The following two results of Allison describe Lie algebras which have such decomposition properties in terms of Jternary and structurable algebras. The definition of J-ternary algebra can be found in (6.2) below and of structurable algebra in (7.6). In stating these results we use the notion of an 𝔰𝔩_{2} -triple to mean a triple of elements e, f, h which satisfy the standard 𝔰𝔩_{2} -relations, [e, f] = h, [h, e] = 2e, and [h, f] = −2f.

Proposition 2.4. [2, Thm. 1] (see also [11]) Assume ℒ is a Lie algebra which contains a subalgebra 𝒯 = ⟨e, f, h⟩ generated by an 𝔰𝔩_{2} -triple such that 𝒯 centralizes no proper ideals of ℒ. Assume ℒ as an 𝒯 -module is the direct sum of irreducible 𝒯 -modules of dimensions 1, 2, 3, and ℒ is generated by the −2, −1, 1, and 2 eigenspaces for ad h. Then ℒ ≅ K(𝔍, 𝔐) = 𝔍̃⊕𝔐̃⊕𝔦𝔫𝔰𝔱𝔯(𝔍, 𝔐)⊕𝔐⊕𝔍, the Tits-Kantor-Koecher Lie algebra of a J-ternary algebra (𝔍, 𝔐).

Proposition 2.5. ([4]) Assume ℒ is a Lie algebra which contains a subalgebra 𝒯 = ⟨e, f, h⟩ generated by an 𝔰𝔩_{2} -triple such that 𝒯 centralizes no proper ideals of ℒ. Assume ℒ as a 𝒯 -module is the direct sum of irreducible 𝒯 -modules of dimensions 1, 3, 5, and ℒ is generated by the −2 and 2 eigenspaces for ad h. Then ℒ ≅ 𝒦(X,^{−}) = S̃⊕ X̃ ⊕ 𝔦𝔫𝔰𝔱𝔯(X,^{−}) ⊕ X ⊕ S, the Tits-Kantor-Koecher Lie algebra of a structurable algebra (X,^{−}) with involution “^{−} ” and skew-hermitian elements S relative to ^{−}.

Using those propositions we may conclude the following about a BC_{1} - graded Lie algebra L modulo its center Z(L).

Theorem 2.6. Let L be a BC_{1}-graded Lie algebra with grading subalgebra 𝔤.

(i) When 𝔤 is of type B_{1} , then L = (𝔤 ⊗ A) ⊕ (𝔰 ⊗ B) ⊕ D and L/Z(L) ≅ K(𝔞, η) = B̃ ⊕ 𝔞̃ ⊕ 𝔦𝔫𝔰𝔱𝔯(𝔞, η) ⊕ 𝔞 ⊕ B, where 𝔞 = A ⊕ B is a structurable algebra with involution η, and A is the set of symmetric elements and B the set of skew-hermitian elements relative to η.

(ii) When 𝔤 is of type C_{1} , then L = (𝔤 ⊗ A) ⊕ (V ⊗ C) ⊕ D and L/Z(L) ≅ K(A, C) = Ã ⊕ C̃ ⊕ 𝔦𝔫𝔰𝔱𝔯(A, C) ⊕ C ⊕ A where (A, C) is a J-ternary algebra.

2.7 Since Z(L) is contained in D, we can use the weight space decomposition of L/Z(L) relative to ad h to realize the isomorphisms in (i) and (ii) more conceptually. Indeed, we have the following identifications:

𝔰_{−2ϵ_{1}} ⊗ B →B̃,  𝔰_{2ϵ_{1}} ⊗ B → B,
(i)  (𝔤_{−ϵ_{1}} ⊗ A) ⊕ (𝔰_{−ϵ_{1}} ⊗ B) → 𝔞̃,  (𝔤_{ϵ_{1}} ⊗ A) ⊕ (𝔰_{ϵ_{1}} ⊗ B) → 𝔞,
(𝔤_{0} ⊗ A) ⊕ (𝔰_{0}⊗B) ⊕ D/Z(L) → 𝔦𝔫𝔰𝔱𝔯(𝔞, η),

𝔤_{−2ϵ_{1}} ⊗ A →Ã,  𝔤_{2ϵ_{1}} ⊗ A → A,
(ii)  V_{−ϵ_{1}} ⊗ C →C̃,  V_{ϵ_{1}} ⊗ C → C,
(𝔤_{0} ⊗ A)⊕D/Z(L) → 𝔦𝔫𝔰𝔱𝔯(A, C).

2.8 In Sections 3 and 4, we study Jordan-Kantor pairs, BC_{1} -graded Lie algebras of type D_{1} , and more generally, 5-graded Lie algebras. Section 5 investigates central extensions of 5-graded Lie algebras culminating in a classification of the BC_{1} -graded Lie algebras of type D_{1} . The derivations and invariant forms of these Lie algebras (in fact of all BC_{1} -graded Lie algebras) are computed in Sections 8 and 9. In Sections 6 and 7, we return to BC_{1} -graded Lie algebras of type B_{1} and C_{1} , J-ternary algebras, and structurable algebras. There we apply the results on BC_{1} -graded Lie algebras of type D_{1} to gain insight into these Lie algebras and determine their classification. This is possible because of the following result which relates BC_{1} -graded Lie algebras of various types. This theorem employs the notion of a grading element in a Z-graded Lie algebra L = ⨁_{i∈Z} L_{i}. By that we mean an element h such that ad h |_{L_{i}}= i Id|_{L_{i}}.

Theorem 2.9. For a Lie algebra L, the following are equivalent:

(i) L = ⨁_{i=−2}^{2} L_{i} is a 5-graded Lie algebra with a grading element h ∈ L_{0} and with L_{0} = [L_{−1}, L_{1}] + [L_{−2}, L_{2}]; (ii) L is a BC_{1} -graded Lie algebra of type D_{1} ; (iii) L is a BC_{1} -graded Lie algebra; (iv) L is Δ-graded Lie algebra.

Moreover, in this case

(a) L = ⨁_{i=−2}^{2} L_{i} is a BC_{1} -graded Lie algebra of type B_{1} if and only if L_{−1}⊕L_{0}⊕L_{1} contains an 𝔰𝔩_{2} -triple ⟨e, f, h⟩ with (1/2)h being a grading element of L.

(b) L = ⨁_{i=−2}^{2} L_{i} is a BC_{1} -graded Lie algebra of type C_{1} if and only if L_{−2} ⊕ L_{0} ⊕ L_{2} contains an 𝔰𝔩_{2} -triple ⟨e, f, h⟩ with h being a grading element of L.

Proof. The equivalence of (i) and (ii) is apparent. For (iii) ⇒ (i) observe that if L is BC_{1} -graded of type B_{1} , then L decomposes into modules for 𝔤 ≅ 𝔰𝔩_{2} = ⟨e, f, h⟩ of dimensions 1, 3, 5. From the representation theory of 𝔰𝔩_{2} we know then that (1/2) ad h has eigenvalues −2, −1, 0, 1, 2 on L. This gives L a 5-grading relative to the grading element (1/2)h. The condition L_{0} = [L_{−1}, L_{1}] + [L_{−2}, L_{2}] is (iii) of Definition 1.7. Therefore, L is BC_{1} -graded of type D_{1} . The case when L is BC_{1} -graded of type C_{1} can be done similarly.

For the proof of (iv) ⇒ (ii), suppose L is a Δ-graded Lie algebra with grading subalgebra 𝔤. If L is a BC_{1} -graded Lie algebra of type D_{1} , the result is evident. Assume otherwise, so that the grading subalgebra 𝔤 is a split finite-dimensional semisimple algebra. In [3, Thm. 10] it is shown that there exists an 𝔰𝔩_{2} -triple ⟨e, f, h⟩ in 𝔤 such that eigenvalues of ad h on 𝔤 belong to {−4, −2, 0, 2, 4}, and the spaces 𝔤_{i}, i ≠ 0, generate 𝔤. (This was done in [3] for 𝔤 simple. But when 𝔤 = 𝔤^{(1)} ⊕ 𝔤^{(2)} is of type D_{2} , then 𝔤^{(i)} ≅ 𝔰𝔩_{2} = ⟨e_{i}, f_{i}, h_{i}⟩, i = 1, 2, and e = e_{1} + e_{2}, f = f_{1} + f_{2}, h = h_{1} + h_{2} have the desired properties.) For this choice of h, the Lie algebra L is graded by the eigenspaces of (1/2) ad h: L = ⨁_{i=−2}^{2} L_{i}. The subalgebra M of L generated by the spaces L_{i}, i ≠ 0 is an ideal of L which contains 𝔤. Now it follows that L ⊆ M and that L is BC_{1} -graded of type D_{1} .

3. Jordan-Kantor pairs

3.1 All vector spaces are assumed to be over a field F of characteristic zero. This restriction on the characteristic is to insure that our BC_{1} -graded algebras have a decomposition into 𝔰𝔩_{2} -modules as in (2.2). However, the results in our paper 98 Benkart and Smirnov

which make no reference to BC_{1} -graded algebras are valid under the weaker assumption that the characteristic of F is not equal to 2 or 3.

Henceforth σ ∈ {−, +} with the obvious interpretation of −σ. Recall that a pair of spaces J = (J_{−}, J_{+}) is said to be a (linear) Jordan pair provided there is a trilinear product {a, b, c} = D_{a,b}c ∈ J_{σ} for a, c ∈ J_{σ} and b ∈ J_{−σ} such that for all d ∈ J_{−σ}, (JP) {a, b, c} = {c, b, a} and [D_{a,b}, D_{c,d}] = D_{D_{a,b}c,d} − D_{c,D_{b,a}d}. The equivalence of this definition with the definition of Jordan pair given by Loos in [28] follows from [28, Prop. 2.2].

Suppose that M = (M_{−}, M_{+}) is a special J -bimodule relative to the action ∘, i.e., a ∘ x ∈ M_{σ} for a ∈ J_{σ}, x ∈ M_{−σ} and (JK1) {a, b, c} ∘ y = a ∘ (b ∘ (c ∘ y)) + c ∘ (b ∘ (a ∘ y)) for all a, c ∈ J_{σ}, b ∈ J_{−σ}, y ∈ M_{−σ}. Assume that M = (M_{−}, M_{+}) possesses a triple product {x, y, z} = V_{x,y}z ∈ M_{σ} for x, z ∈ M_{σ} and y ∈ M_{−σ} such that (JK2) [V_{x,y}, V_{z,w}] = V_{V_{x,y}z,w} − V_{z,V_{y,x}w} for all x, z ∈ M_{σ}, y, w ∈ M_{−σ}. In addition, we suppose that there are anticommutative bilinear maps κ : M_{σ} × M_{σ} → J_{σ} which satisfy the following identities: (JK3) κ(x, z) ∘ y = {x, y, z} − {z, y, x}, (JK4) κ(x, z) ∘ (b ∘ u) = {z, b ∘ x, u} − {x, b ∘ z, u}, (JK5) κ(b ∘ x, y) ∘ z = b ∘ {x, y, z} + {y, x, b ∘ z}, (JK6) {a, b, κ(x, z)} = κ(a ∘ (b ∘ x), z) + κ(x, a ∘ (b ∘ z)), (JK7) {a, κ(y, w), c} = κ(a ∘ w, c ∘ y) + κ(c ∘ w, a ∘ y), (JK8) κ(κ(z, u) ∘ y, x) = κ({x, y, z}, u) + κ(z, {x, y, u}), for all x, z, u ∈ M_{σ}, y, w ∈ M_{−σ}, a, c ∈ J_{σ}, b ∈ J_{−σ}.

A pair (J, M) satisfying (JP) and (JK1-JK8) is said to be a Jordan-Kantor pair (or simply a JKP for short).

Remark 3.2. As we will see in Section 7, the pair M = (M_{−}, M_{+}) with its products is a Kantor pair. Thus, any JKP (J, M) consists of a Jordan pair J and a Kantor pair M related in a certain way. Later we will discuss how structurable algebras, Kantor pairs, and J-ternary algebras all can be regarded as JKPs. 3.3 The following useful identity can be derived from relations (JK3-5): (JK9) a∘(b∘{x, y, z}) = {a∘(b∘x), y, z}−{x, b∘(a∘y), z}+{x, y, a∘(b∘z)}.

To see this, first observe that (JK5) and (JK3) give b ∘ {x, y, z} = {b ∘ x, z, y} − {y, z, b ∘ x} − {y, x, b ∘ z}. Applying this identity repeatedly shows that

a ∘ (b ∘ {x, y, z}) = a ∘ {b ∘ x, z, y} − a ∘ {y, z, b ∘ x} − a ∘ {y, x, b ∘ z}
= {a ∘ (b ∘ x), y, z} − {z, y, a ∘ (b ∘ x)} − {z, b ∘ x, a ∘ y}
− {a ∘ y, b ∘ x, z} + {z, b ∘ x, a ∘ y} + {z, y, a ∘ (b ∘ x)}
− {a ∘ y, b ∘ z, x} + {x, b ∘ z, a ∘ y} + {x, y, a ∘ (b ∘ z)}
= {a ∘ (b ∘ x), y, z} − [{a ∘ y, b ∘ x, z} + {a ∘ y, b ∘ z, x}
− {x, b ∘ z, a ∘ y}] + {x, y, a ∘ (b ∘ z)}.

Finally, note that {a∘y, b∘x, z}+{a∘y, b∘z, x}−{x, b∘z, a∘y} = {x, b∘(a∘y), z} according to (JK4) and (JK3).

The structure and inner structure algebras of a JKP

3.4 We construct a Lie algebra out of a Jordan-Kantor pair following the usual pattern of first constructing the structure and inner structure algebras. They are Lie subalgebras of the Lie algebra

ℰ(J, M) = End(J_{−}) ⊕ End(M_{−}) ⊕ End(M_{+}) ⊕ End(J_{+})

under the commutator product [T, T′] = T T′ − T′T. It has been customary to write (S, S^{ϵ}) for elements of End(J_{−}) ⊕ End(J_{+}). Here we adopt a nonconventional but convenient notation T for a pair (S, S^{ϵ}). Then S = T |_{J_{−}} and S^{ϵ} = T |_{J_{+}} are the restrictions of T to J_{−} and J_{+} respectively. In other words, we identify ℰ(J, M) = End(J_{−}) ⊕ End(M_{−}) ⊕ End(M_{+}) ⊕ End(J_{+}) with its natural diagonal embedding into End(J_{−} ⊕ M_{−} ⊕ M_{+} ⊕ J_{+}).

3.5 Define the structure algebra 𝔰𝔱𝔯(J, M) of the JKP (J, M) to be the set of all endomorphisms T ∈ ℰ(J, M) such that

(STR1) [T, D_{a,b}] = D_{T a,b} + D_{a,T b}, [T, V_{x,y}] = V_{T x,y} + V_{x,T y},

(STR2) T (a ∘ y) = T a ∘ y + a ∘ T y, T κ(x, z) = κ(T x, z) + κ(x, T z),

for all x, z ∈ M_{σ}, y ∈ M_{−σ}, a ∈ J_{σ}, b ∈ J_{−σ}. It is easy to see that 𝔰𝔱𝔯(J, M) is closed under the commutator product and hence is a Lie subalgebra of ℰ(J, M).

For a ∈ J_{+}, b ∈ J_{−}, there is a unique element δ(a, b) of ℰ(J, M) such

that

\begin{aligned}δ(a, b)c & = \{a, b, c\}, δ(a, b)d = −\{b, a, d\}, \\ δ(a, b)x & = a ∘ (b ∘ x), δ(a, b)y = −b ∘ (a ∘ y), \end{aligned}

for all c ∈ J_{+}, d ∈ J_{−}, x, ∈ M_{+}, y ∈ M_{−}. For convenience, we put δ(b, a) = −δ(a, b). This eliminates the (+/–)-asymmetry in the formulas above and allows them to be written as

(3.6)  δ(a, b)c = {a, b, c} and δ(a, b)x = a ∘ (b ∘ x)

for all a, c ∈ J_{σ}, b ∈ J_{−σ} and x ∈ M_{σ}.

Similarly, for x ∈ M_{+}, y ∈ M_{−} we define υ(x, y) of ℰ(J, M) by

\begin{aligned}υ(x, y)z & = \{x, y, z\}, υ(x, y)w = −\{y, x, w\}, \\ υ(x, y)a & = κ(a ∘ y, x), υ(x, y)b = −κ(b ∘ x, y), \end{aligned}

for z ∈ M_{+}, w ∈ M_{−}, a ∈ J_{+}, b ∈ J_{−}. Again, by setting υ(y, x) = −υ(x, y) we have

(3.7)  υ(x, y)z = {x, y, z} and υ(x, y)a = κ(a ∘ y, x)

for all x, z ∈ M_{σ}, y, w ∈ M_{−σ}, a ∈ J_{σ}, b ∈ J_{−σ}.

Lemma 3.8. The endomorphisms δ(a, b), υ(x, y) belong to 𝔰𝔱𝔯(J, M) for all a ∈ J_{σ}, b ∈ J_{−σ}, x ∈ M_{σ}, y ∈ M_{−σ}. Proof. We need to verify that the identities (STR1–STR2) are satisfied for these endomorphisms. We begin with δ(a, b).

Since δ(a, b)|_{J_{σ}} = D_{a,b}, the first part of (STR1) is merely the second identity in (JP). The second part is equivalent to (JK9). Equation (JK1) is equivalent to the first equation in (STR2) and (JK6) to the second equation in (STR2) for T = δ(a, b).

Now when T = υ(x, y), the second part of (STR1) is exactly (JK2). The first identity in (STR2) for υ(x, y) is equivalent to (JK5) and the second to (JK8). The only identity which requires some calculation is the first part of (STR1). For any c ∈ J_{σ}, [υ(x, y), D_{a,b}](c) = D_{υ(x,y)a,b}(c) + D_{a,υ(x,y)b}(c) is equivalent to saying

(JK7′) κ({a, b, c}∘y, x) = {κ(a∘y, x), b, c}−{a, κ(b∘x, y), c}+{a, b, κ(c∘y, x)}.

By (JK1), that relation is equivalent to

\begin{aligned}κ(a ∘ (b ∘ (c ∘ y)), x) & + κ(c ∘ (b ∘ (a ∘ y)), x) \\ & = \{κ(a ∘ y, x), b, c\} − \{a, κ(b ∘ x, y), c\} + \{a, b, κ(c ∘ y, x)\}, \end{aligned}

which is equivalent to

{a, κ(b ∘ x, y), c} = κ(c ∘ y, a ∘ (b ∘ x)) + k(a ∘ y, c ∘ (b ∘ x))

by (JK6). This is exactly (JK7) with b ∘ x as w.*

Lemma 3.9. If D ∈ 𝔰𝔱𝔯(J, M), then

[D, δ(a, b)] = δ(Da, b) + δ(a, Db) [D, υ(x, y)] = υ(Dx, y) + υ(x, Dy)

for any a ∈ J_{σ}, b ∈ J_{−σ}, x ∈ M_{σ}, y ∈ M_{−σ}. Proof. This follows directly from (STR1-STR2) and the definitions of δ(a, b) and υ(x, y).

* If J_{−σ}M_{σ} = M_{−σ} holds, then relation (JK7′) is equivalent to (JK7). In this case, the two conditions (STR1) and (STR2) are equivalent to (JK1-2) and (JK9), (JK5-8), and the last part of (JP).

3.10 The subalgebra 𝔦𝔫𝔰𝔱𝔯(J, M) of 𝔰𝔱𝔯(J, M) generated by the elements δ(a, b) and υ(x, y) is said to be the inner structure algebra of the JKP (J, M). Lemma 3.9 states that the subspaces δ(J_{+}, J_{−}) and υ(M_{+}, M_{−}) are ideals of the Lie algebra 𝔰𝔱𝔯(J, M), and that

(3.11)  𝔦𝔫𝔰𝔱𝔯(J, M) = δ(J_{+}, J_{−}) + υ(M_{+}, M_{−}).

The structure algebra 𝔰𝔱𝔯(J, M) also contains the element h = (−2 Id, − Id, Id, 2 Id) ∈ ℰ(J, M) = End(J_{−}) ⊕ End(M_{−}) ⊕ End(M_{+}) ⊕ End(J_{+}). The endomorphism h lies in the center of 𝔰𝔱𝔯(J, M) and is called the grading derivation of (J, M).

Invariant forms of Jordan-Kantor pairs

3.12 Let P = (J, M) be a JKP. An invariant form on P consists of a pair of bilinear forms ( , ) : J_{+} × J_{−} → F, ( , ) : M_{+} × M_{−} → F for which the following identities hold: (IF1) (D_{a,b}c, d) = (c, D_{b,a}d), (IF2) (V_{x,y}z, w) = (z, V_{y,x}w), (IF3) (κ(x, z), b) = (z, b ∘ x), (IF4) (a, κ(y, w)) = (w, a ∘ y), for all a, c ∈ J_{+} , b, d ∈ J_{−}, x, z ∈ M_{+} , y, w ∈ M_{−}. We extend the form by setting (b, a) = (a, b) and (y, x) = (x, y) for a ∈ J_{+} , b ∈ J_{−}, x ∈ M_{+} , and y ∈ M_{−}. It follows from (IF3) and κ(x, z) = −κ(z, x) that (IF5) (x, b ∘ z) = −(z, b ∘ x).

3.13 We note that this definition implies that δ(a, b) and υ(x, y) are skewsymmetric transformations on the vector space J_{−} ⊕M_{−} ⊕M_{+} ⊕J_{+} with respect to the symmetric bilinear form g defined by

g(p, q) = \{ \begin{matrix}(p, q) & if p ∈ J_{σ} and q ∈ J_{−σ}, \\ & or if p ∈ M_{σ} and q ∈ M_{−σ}, \\ 0 & otherwise \end{matrix}

Often we identify g with the invariant form on (J, M).

4. The Lie algebra 𝔏(J, M, 𝔇)

4.1 Let (J, M) be a JKP and assume 𝔇 is a Lie subalgebra of 𝔰𝔱𝔯(J, M) which contains 𝔦𝔫𝔰𝔱𝔯(J, M). On the space

𝔏(J, M, 𝔇) = J_{−} ⊕ M_{−} ⊕ 𝔇 ⊕ M_{+} ⊕ J_{+}

define an anticommutative product [ , ] by the following rules:

(4.2)

[D, D′] is the product in 𝔇, [D, (x, a)_{σ}] = (D(x), D(a))_{σ}, [(x, a)_{σ}, (z, c)_{σ}] = (0, κ(x, z))_{σ}, [(x, a)_{σ}, (y, b)_{−σ}] = (−b ∘ x, 0)_{−σ} + δ(a, b) + υ(x, y) + (a ∘ y, 0)_{σ},

for all pairs (x, a)_{σ}, (z, c)_{σ} ∈ M_{σ} ⊕ J_{σ}, (y, b)_{−σ} ∈ M_{−σ} ⊕ J_{−σ}, and D, D′ ∈ 𝔇.

Theorem 4.3. The space 𝔏 = 𝔏(J, M, 𝔇) with the product in (4.2) is a Lie algebra. It has a 5-grading 𝔏 = 𝔏_{−2} ⊕ 𝔏_{−1} ⊕ 𝔏_{0} ⊕ 𝔏_{1} ⊕ 𝔏_{2} , where 𝔏_{−2} = J_{−}, 𝔏_{−1} = M_{−}, 𝔏_{0} = 𝔇, 𝔏_{1} = M_{+} , 𝔏_{2} = J_{+} , and ker(ad_{#}) = 0 where the map ad_{#} : 𝔏_{0} → ℰ(J, M) = End(𝔏_{−2}) ⊕ End(𝔏_{−1}) ⊕ End(𝔏_{1}) ⊕ End(𝔏_{2}) is defined by ad_{#}(l_{0}) = ad(l_{0})|_{𝔏_{−2}⊕𝔏_{−1}⊕𝔏_{1}⊕𝔏_{2}} for all l_{0} ∈ 𝔏_{0} . Proof. To prove the first assertion it suffices to verify the Jacobi identity 𝔧(X_{1}, X_{2}, X_{3}) = [[X_{1}, X_{2}], X_{3}] + [[X_{2}, X_{3}], X_{1}] + [[X_{3}, X_{1}], X_{2}] = 0 for arbitrary elements X_{1}, X_{2}, X_{3} ∈ (M_{σ} ⊕ J_{σ}) ⋃ 𝔇 ⋃(M_{−σ} ⊕ J_{−σ}).

If two or more of these elements come from 𝔇, the identity is true because 𝔇 is a Lie algebra of endomorphisms.

If X_{1} = (x, a)_{σ}, X_{2} = (z, c)_{σ} ∈ M_{σ} ⊕ J_{σ} and X_{3} = D ∈ 𝔇,

𝔧(X_{1}, X_{2}, X_{3}) = −(0, D(κ(x, z)))_{σ} + (0, κ(D(z), x))_{σ} + (0, κ(D(x), z))_{σ} = 0

because of (STR2) and the anticommutativity of κ.

For X_{1} = (x, a)_{σ} ∈ M_{σ} ⊕ J_{σ}, X_{2} = (y, b)_{−σ} ∈ M_{−σ} ⊕ J_{−σ}, and X_{3} = D ∈ 𝔇,

𝔧(X_{1}, X_{2}, X_{3}) = (D(b ∘ x) − D(b) ∘ x − b ∘ D(x), 0)_{−σ}
+ ([δ(a, b), D] + δ(D(a), b) + δ(a, D(b))
+ [υ(x, y), D] + υ(D(x), y) + υ(x, D(y)))
+ ( − D(a ∘ y) + D(a) ∘ y + a ∘ D(y), 0)_{σ} = 0

due to (STR2) and Lemma 3.9.

Finally, if X_{1} = (x, a)_{σ}, X_{3} = (z, c)_{σ} ∈ M_{σ} ⊕ J_{σ}, and X_{2} = (y, b)_{−σ} ∈ M_{−σ} ⊕ J_{−σ}, then

𝔧(X_{1}, X_{2}, X_{3}) = υ(z, b ∘ x) − υ(x, b ∘ z) + δ(κ(z, x), b)
+ (c ∘ (b ∘ x) − δ(c, b)x + δ(a, b)z − a ∘ (b ∘ z)
+ υ(x, y)z − υ(z, y)x + κ(z, x) ∘ y, δ(a, b)c − δ(c, b)a
+ υ(x, y)c − κ(c ∘ y, x) + κ(a ∘ y, z) − υ(z, y)a)_{σ} = 0.

Here we have made use of the relations δ(a, b)c−δ(c, b)a = 0, c∘(b∘x)−δ(c, b)x = 0, δ(a, b)z −a∘(b∘z) = 0, υ(x, y)c−κ(c∘y, x) = 0, and κ(a∘y, z)−υ(z, y)a = 0 which follow from the definitions of δ and υ, and of the equation υ(x, y)z − υ(z, y)x + κ(z, x) ∘ y = 0 which is (JK4). The relation

(4.4)  υ(z, b ∘ x) − υ(x, b ∘ z) + δ(κ(z, x), b) = 0,

which is needed for the remaining terms, is equivalent to (JK5). Thus, we have checked the Jacobi identity for all possible choices of X_{1} , X_{2} , X_{3} .

The assertion about the grading of 𝔏 follows immediately from (4.2). The last statement of the theorem follows from the fact that elements of 𝔇 are endomorphisms and from the formulas in (4.2).

It is convenient in what follows to adopt the following conventions: if P = (J, M) is a Jordan-Kantor pair, then

\begin{matrix}𝔏(P, 𝔇) := & 𝔏(J, M, 𝔇) \\ 𝔏(P) (or 𝔏(J, M)) := & 𝔏(J, M, 𝔇) when 𝔇 = 𝔦𝔫𝔰𝔱𝔯(J, M). \end{matrix}

A converse of the last theorem is the essence of

Theorem 4.5. Let L be a Lie algebra with a 5-grading L = L_{−2} ⊕ L_{−1} ⊕ L_{0} ⊕ L_{1} ⊕ L_{2} . Then

(i) J = (L_{−2}, L_{2}) is a Jordan pair with the respect to the product {a, b, c} = [[a, b], c];

(ii) M = (L_{−1}, L_{1}) is a special J -bimodule under the action a ∘ x = [a, x]; (iii) the pair (J, M) together with the triple product {x, y, z} = [[x, y], z] and

the map κ(x, z) = [x, z] form a JKP denoted by 𝒫(L); (iv) if ker(ad_{#}) = 0 where the map ad_{#} : L_{0} → ℰ(J, M) = End(L_{−2}) ⊕

End(L_{−1}) ⊕ End(L_{1}) ⊕ End(L_{2}) is defined by

ad_{#} l_{0} = ad l_{0} |_{L_{−2}⊕L_{−1}⊕L_{1}⊕L_{2}}

for l_{0} ∈ L_{0} , then L is isomorphic to the Lie algebra 𝔏(𝒫(L), 𝔇) for some Lie subalgebra 𝔇 of 𝔰𝔱𝔯(𝒫(L)) which contains 𝔦𝔫𝔰𝔱𝔯(𝒫(L)). If in addition L_{0} = [L_{−1}, L_{1}] + [L_{−2}, L_{2}], then 𝔇 = 𝔦𝔫𝔰𝔱𝔯(𝒫(L)).

Proof. Assertions (i) and (ii) follow directly from the Jacobi identity and properties of the grading. For example, if a, c ∈ L_{2σ}, b ∈ L_{−2σ}, x ∈ L_{−σ}, then [[[a, b], c], x] = [a, [b, [c, x]]] + [c, [b, [a, x]]], which shows that (JK1) holds.

Identities (JK2-JK8) can be readily verified using the triple product {x, y, z} = [[x, y], z], the map κ(x, z) = [x, z], and the Jacobi identity for L. For example, if x, z ∈ L_{σ} and y ∈ L_{−σ} then κ(x, z) ∘ y = [[x, z], y] = [[x, y], z] − [[z, y], x] = {x, y, z} − {z, y, x} justifying identity (JK3). Thus, the pair (J, M) is a JKP.

Assume now ker(ad_{#}) = 0. We argue that L_{0} is isomorphic to some subalgebra 𝔇 of 𝔰𝔱𝔯(J, M) which contains 𝔦𝔫𝔰𝔱𝔯(J, M) and prove that L is isomorphic to 𝔏(J, M, 𝔇). The adjoint map ad_{#} : L_{0} → ℰ(J, M) = End(L_{−2}) ⊕ End(L_{−1}) ⊕ End(L_{1}) ⊕ End(L_{2}) is a Lie algebra monomorphism. Denote the image of L_{0} in ℰ(J, M) by 𝔇. The presence of the Jacobi identity for L implies that identities (STR1) and (STR2) hold. It is easy to check that ad [a, b] = δ(a, b) and ad [x, y] = υ(x, y) for any a ∈ L_{2σ}, b ∈ L_{−2σ}, x ∈ L_{σ}, y ∈ L_{−σ}. Thus, 𝔇 contains 𝔦𝔫𝔰𝔱𝔯(J, M). Now the fact that L is isomorphic to the Lie algebra 𝔏(J, M, 𝔇) follows from the definitions of the products in (J, M) and 𝔏(J, M, 𝔇).

Suppose in addition that L_{0} = [L_{−1}, L_{1}] + [L_{−2}, L_{2}]. Then 𝔇 = ad_{#} L_{0} = [ad L_{−1}, ad L_{1}] + [ad L_{−2}, ad L_{2}] = 𝔦𝔫𝔰𝔱𝔯(𝒫(L)).

Corollary 4.6. A Lie algebra L is isomorphic to a Lie algebra 𝔏(J, M, 𝔇) for a JKP (J, M) and a subalgebra 𝔇 of 𝔰𝔱𝔯(J, M) which contains 𝔦𝔫𝔰𝔱𝔯(J, M) if and only if L is a 5-graded Lie algebra, L = L_{−2} ⊕ L_{−1} ⊕ L_{0} ⊕ L_{1} ⊕ L_{2} , with ker(ad_{#}) = 0.

Now for every 5-graded Lie algebra L we can construct the JKP 𝒫(L), and for every JKP P we can construct the 5-graded Lie algebra 𝔏(P). To compare these constructions we require the notion of morphisms of JKPs.

Definition 4.7. Let P = (J, M) and P′ = (J′, M′) be two Jordan-Kantor pairs. A linear map ρ : J_{−} ⊕ M_{−} ⊕ M_{+} ⊕ J_{+} → J′_{−} ⊕ M′_{−} ⊕ M′_{+} ⊕ J′_{+} , ρ(x) = x′, is said to be a JKP homomorphism if ρ(J_{σ}) ⊆ J′_{σ}, ρ(M_{σ}) ⊆ M′_{σ}, and

\begin{matrix}\{a, b, c\}′ = \{a′, b′, c′\}, & \{x, y, z\}′ = \{x′, y′, z′\}, \\ (a ∘ x)′ = a′ ∘ x′, & κ(x, z)′ = κ(x′, z′) \end{matrix}

for all a, c ∈ J_{σ}, b ∈ J_{−σ}, x, z ∈ M_{σ}, and y ∈ M_{−σ}.

We say that ρ is an isomorphism if it is bijective, and in this case, write P ≅ P′.

Lemma 4.8.

(i) Let L and L′ be 5-graded Lie algebras. If L ≅ L′, then 𝒫(L) ≅ 𝒫(L′).

(ii) Let P and P′ be JKPs. If P ≅ P′, then 𝔏(P) ≅ 𝔏(P′).

(iii) For every JKP P one has 𝒫(𝔏(P)) = P.

(iv) Let L be a 5-graded Lie algebra. Then 𝔏(𝒫(L)) ≅ L if and only if L ≅ 𝔏(P) for some JKP P.

Proof. Statements (i) and (ii) are immediate. Assertion (iii) follows from (4.2) and (i)-(iii) of Theorem 4.5.

If L ≅ 𝔏(P), then from (i)-(iii) of this lemma we have that 𝔏(𝒫(L)) ≅ 𝔏(𝒫(𝔏(P))) ≅ 𝔏(P) ≅ L. The converse is obvious.

Recall that an element h of a Z-graded Lie algebra L = ⨁_{i∈Z} L_{i} is called a grading element if ad h|_{L_{i}} = i Id|_{L_{i}} for every i ∈ Z. It is unique up to a central element. In this terminology, the grading derivation of 𝔰𝔱𝔯(J, M) in (3.10) is a grading element of the Lie algebra 𝔏(J, M, 𝔇).

Theorem 4.9. Every BC_{1} -graded algebra L of type D_{1} is a central extension of a Lie algebra 𝔏(P) for a JKP P such that 𝔦𝔫𝔰𝔱𝔯(P) contains a grading derivation.

Proof. A BC_{1} -graded algebra L of type D_{1} is 5-graded, L = L_{−2}⊕L_{−1}⊕L_{0}⊕ L_{1} ⊕ L_{2} relative to 𝔥 = Fh, and L_{0} = [L_{−1}, L_{1}] + [L_{−2}, L_{2}] contains the grading element h. The latter fact implies that Z(L) ⊆ L_{0} . Thus by Theorem 4.5, the algebra \overline{L} = L/Z(L) is isomorphic to 𝔏 = 𝔏(P, 𝔇) where P = 𝒫(\overline{L}) and 𝔇 is some subalgebra of 𝔰𝔱𝔯(P) which contains 𝔦𝔫𝔰𝔱𝔯(P). Clearly the coset containing h is a grading element in \overline{L}. Moreover, 𝔇 ⊆ 𝔏_{0} ⊆ [𝔏_{−1}, 𝔏_{1}] + [𝔏_{−2}, 𝔏_{2}] ⊆ 𝔦𝔫𝔰𝔱𝔯(P), so 𝔏(P, 𝔇) = 𝔏(P).

Lemma 4.10. Let L = L_{−2} ⊕L_{−1} ⊕L_{0} ⊕L_{1} ⊕L_{2} be a 5-graded Lie algebra with an invariant symmetric bilinear form f̄ such that f̄(L_{i}, L_{j}) = 0 if i + j ≠ 0.

Then the restriction of f̄ to P = 𝒫(L) is an invariant form of the JKP P, and the restriction of f̄ to L_{0} is an invariant symmetric bilinear form of the Lie algebra L_{0} .

Conversely, if f is an invariant form of the JKP P = 𝒫(L) and f_{0} is an invariant symmetric bilinear form of the Lie algebra L_{0} such that

(4.11)  f_{0}([l_{−σ}, l_{σ}], l_{0}) = f(l_{−σ}, [l_{σ}, l_{0}])

for all l_{0} ∈ L_{0} and (l_{−σ}, l_{σ}) ∈ (L_{−σ}, L_{σ}) ⋃(L_{−2σ}, L_{2σ}), then the map f̄ defined by

f̄(p + l_{0}, p′ + l′_{0}) = f(p, p′) + f_{0}(l_{0}, l′_{0})

for p, p′ ∈ P and l_{0}, l′_{0} ∈ L_{0} is an invariant symmetric bilinear form of the Lie algebra L.

Proof. The first part of the lemma is an immediate consequence of the Jacobi identity and the invariance of f.

For the second part one has to check that the equation

f̄([l_{i}, l_{j}], l_{k}) = f̄([l_{i}, [l_{j}, l_{k}])

holds for l_{i} ∈ L_{i}, l_{j} ∈ L_{j}, l_{k} ∈ L_{k}. Both sides of this relation are equal to zero when one of i, j, k is zero, and the sum of other two is nonzero. If one of i, j, k is zero, and the sum of other two is zero, the equation follows from (4.11). If i, j, k are all nonzero, or all zero, it follows from properties of f and f_{0} .

5. Central extensions of 𝔏(J, M) and

the classification of BC_{1} -graded Lie algebras of type D_{1}

5.1 The aim of this section is to describe the central extensions of the Lie algebra 𝔏(J, M) for P = (J, M) a JKP. Recall that 𝔏(J, M) or 𝔏(P) is our shorthand for the Lie algebra 𝔏(J, M, 𝔦𝔫𝔰𝔱𝔯(J, M)) in (4.1). Our results here are adaptations of ideas from [10], [21], and [7] to our particular setting.

Full skew-dihedral homology

5.2 First we construct the Lie algebra {P, P }. The space {P, P } is the direct sum (J_{+} ⊗ J_{−}) ⨁(M_{+} ⊗ M_{−}) modulo the subspace 𝒰 spanned by the elements

(S1) {a, b, c} ⊗ d − c ⊗ {b, a, d} + {c, d, a} ⊗ b − a ⊗ {d, c, b},

(S2) {x, y, z} ⊗ w − z ⊗ {y, x, w} + {z, w, x} ⊗ y − x ⊗ {w, z, y},

(S3) κ(z, x) ⊗ b − x ⊗ (b ∘ z) + z ⊗ (b ∘ x),

(S4) a ⊗ κ(y, w) − (a ∘ y) ⊗ w + (a ∘ w) ⊗ y.

Here x, z ∈ M_{+}, y, w ∈ M_{−}, a, c ∈ J_{+}, b, d ∈ J_{−}. Let {a, b} and {x, y} denote the cosets a ⊗ b + 𝒰 and x ⊗ y + 𝒰 respectively in {P, P }. Then, {P, P } = {J_{+}, J_{−}} + {M_{+}, M_{−}}. If we add (S3) with z = a ∘ y to (S4) with w = b ∘ x, we obtain another element of 𝒰, (S5) κ(a ∘ y, x) ⊗ b − a ⊗ κ(b ∘ x, y) + a ∘ (b ∘ x) ⊗ y − x ⊗ b ∘ (a ∘ y). 5.3 Since J_{+} , J_{−}, M_{+} , and M_{−} are 𝔰𝔱𝔯(J, M) – modules, the space (J_{+} ⊗ J_{−}) ⊕ (M_{+} ⊗ M_{−}) carries a natural 𝔰𝔱𝔯(J, M)-module structure given by:

T ( ∑_{i} a_{i}⊗b_{i}+∑_{j} x_{j} ⊗y_{j}) = ∑_{i} T a_{i}⊗b_{i}+∑_{i} a_{i}⊗T b_{i}+∑_{j} T x_{j} ⊗y_{j} +∑_{j} x_{j} ⊗T y_{j}

for any T ∈ 𝔰𝔱𝔯(J, M). Thus, the elements in (S1) (S2) and (S5) can be rewritten as (S1) δ(a, b)(c ⊗ d) + δ(c, d)(a ⊗ b), (S2) υ(x, y)(z ⊗ w) + υ(z, w)(x ⊗ y), (S5) υ(x, y)(a ⊗ b) + δ(a, b)(x ⊗ y).

Lemma 5.4. The subspace 𝒰 is invariant under the action of 𝔰𝔱𝔯(J, M). Proof. It suffices to note that the images of the elements in (S1-S4) under the action of 𝔰𝔱𝔯(J, M) are in 𝒰. If T ∈ 𝔰𝔱𝔯(J, M), then T (δ(a, b)(c ⊗ d)) + T (δ(c, d)(a ⊗ b)) = [T, δ(a, b)](c ⊗ d) + δ(a, b)(T (c ⊗ d)) + [T, δ(c, d)](a ⊗ b) + δ(c, d)(T (a ⊗ b)) = δ(T a, b)(c ⊗ d) + δ(a, T b)(c ⊗ d) + δ(a, b)(T c ⊗ d) + δ(a, b)(c ⊗ T d)

+ δ(T c, d)(a ⊗ b) + δ(c, T d)(a ⊗ b) + δ(c, d)(T a ⊗ b) + δ(c, d)(a ⊗ T b) = δ(T a, b)(c ⊗ d) + δ(c, d)(T a ⊗ b) + δ(a, T b)(c ⊗ d) + δ(c, d)(a ⊗ T b)

+ δ(a, b)(T c ⊗ d) + δ(T c, d)(a ⊗ b) + δ(a, b)(c ⊗ T d) + δ(c, T d)(a ⊗ b) ∈ 𝒰;

T (κ(z, x) ⊗ b)−T (x ⊗ b ∘ z) + T (z ⊗ b ∘ x)

= T (κ(z, x)) ⊗ b + κ(z, x) ⊗ T b − T x ⊗ b ∘ z − x ⊗ T (b ∘ z)

+ T z ⊗ b ∘ x + z ⊗ T (b ∘ x)
= κ(T z, x) ⊗ b + κ(z, T x) ⊗ b + κ(z, x) ⊗ T b
− T x ⊗ b ∘ z − x ⊗ T b ∘ z − x ⊗ b ∘ T z

+ T z ⊗ b ∘ x + z ⊗ T b ∘ x + z ⊗ b ∘ T x ∈ 𝒰 .

Similarly one can check that the images of (S2) and (S4) under T are in 𝒰.

5.5 Lemma 5.4 implies that the space {P, P } is a 𝔰𝔱𝔯(J, M)-module satisfying the following identities: (P1) δ(a, b){c, d} = −δ(c, d){a, b}, (P2) δ(a, b){x, y} = −υ(x, y){a, b}, (P3) υ(x, y){z, w} = −υ(z, w){x, y}, (P4) {κ(z, x), b} = {x, b ∘ z} − {z, b ∘ x}, (P5) {a, κ(y, w)} = {a ∘ y, w} − {a ∘ w, y}, (P6) T {a, b} = {T a, b} + {a, T b}, (P7) T {x, y} = {T x, y} + {x, T y}, for all x, z ∈ M_{+}, y, w ∈ M_{−}, a, c ∈ J_{+}, b, d ∈ J_{−}, and T ∈ 𝔰𝔱𝔯(J, M).

The next result will enable us to define a Lie product on {P, P }.

Lemma 5.6. Let L be a Lie algebra and Q be an L-module. Assume that there is a linear map λ : Q → L such that for all x, y ∈ Q

(i) λ(x)y = −λ(y)x,, and (ii) [λ(x), λ(y)] = λ(λ(x)y). Then, Q forms a Lie algebra under the product

[x, y] = λ(x)y. Moreover, λ is a Lie algebra homomorphism. Proof. Anticommutativity of the product follows from (i) and the Jacobi identity from (ii). The assertion about λ being a homomorphism follows from (ii) also.

Remark 5.7. If under the conditions of Lemma 5.6, the space Q possesses a symmetric bilinear form f, and L is represented by skew-symmetric transformations relative to that form, then f is an invariant form of the Lie algebra Q: f([x, y], z) = −f(λ(y)x, z) = f(x, λ(y)z) = f(x, [y, z]).

Proposition 5.8. There is a unique linear map λ : {P, P } → 𝔦𝔫𝔰𝔱𝔯(J, M) such that λ{a, b} = δ(a, b) and λ{x, y} = υ(x, y). The space {P, P } with the product [p, q] = λ(p)q is a Lie algebra which satisfies these relations:

(5.9)  \begin{matrix}[\{a, b\}, \{c, d\}] = \{δ(a, b)c, d\} + \{c, δ(a, b)d\}, \\ [\{a, b\}, \{x, y\}] = \{δ(a, b)x, y\} + \{x, δ(a, b)y\}, \\ [\{x, y\}, \{z, w\}] = \{υ(x, y)z, w\} + \{z, υ(x, y)w\}, \end{matrix}

for a, c ∈ J_{+}, b, d ∈ J_{−} x, z ∈ M_{+}, y, w ∈ M_{−}. Moreover, λ is a Lie algebra homomorphism. Proof. There is a unique map μ : (J_{+} ⊗ J_{−}) ⨁(M_{+} ⊗ M_{−}) → 𝔦𝔫𝔰𝔱𝔯(J, M) such that μ(a ⊗ b) = δ(a, b) and μ(x ⊗ y) = υ(x, y). We claim that μ(𝒰) = 0. Indeed,

μ(δ(a, b)c ⊗ d + c ⊗ δ(a, b)d + δ(c, d)a ⊗ b + a ⊗ δ(c, d)b)
= δ(δ(a, b)c, d) + δ(c, δ(a, b)d) + δ(δ(c, d)a, b) + δ(a, δ(c, d)b)
= [δ(a, b), δ(c, d)] + [δ(c, d), δ(a, b)] = 0 , and

\begin{aligned}μ(κ(z, x) ⊗ b & − x ⊗ b ∘ z + z ⊗ b ∘ x) \\ & = δ(κ(z, x), b) − υ(x, b ∘ z) + υ(z, b ∘ x) = 0 by (4.4) . \end{aligned}

In the same manner one can show that μ sends (S2) and (S4) to zero. Thus, μ(𝒰) = 0, and we have a well-defined map λ : {P, P } → 𝔦𝔫𝔰𝔱𝔯(J, M), λ(p + 𝒰) = μ(p) for which λ{a, b} = δ(a, b) and λ{x, y} = υ(x, y). Next, we verify that λ satisfies the conditions of Lemma 5.6. First,

λ({a, b}){c, d} = δ(a, b){c, d} = −δ(c, d){a, b} = −λ({c, d}){a, b}

by (P1). Analogously, (P2) and (P3) imply λ({a, b}){x, y} = −λ({x, y}){a, b} and λ({x, y}){z, w} = −λ({z, w}){x, y}. Since, the elements {a, b} and {x, y} span {P, P }, (i) of Lemma 5.6 holds in {P, P }. Next we have

[λ({a, b}), λ({c, d})] = [δ(a, b), δ(c, d)] = δ(δ(a, b)c, d) + δ(c, δ(a, b)d)
= λ({δ(a, b)c, d} + {c, δ(a, b)d})
= λ(δ(a, b){c, d}) = λ(λ({a, b}){c, d}),

because of Lemma 3.9 and (P5). Similarly,

[λ({a, b}), λ({x, y})] = λ(λ({a, b}){x, y})

and [λ({x, y}), λ({z, w})] = λ(λ({x, y}){z, w}). Therefore, (ii) of Lemma 5.6 is true for {P, P }, and the result follows from that lemma.

5.10 In concluding our investigations on {P, P }, we note that

\begin{aligned}ker(λ) & = \{∑_{i}\{a_{i}, b_{i}\} + ∑_{j}\{x_{j}, y_{j}\} | ∑_{i} δ(a_{i}, b_{i}) + ∑_{j} υ(x_{j}, y_{j}) = 0 \\ & for a_{i} ∈ J_{+}, b_{i} ∈ J_{−}, x_{j} ∈ M_{+}, y_{j} ∈ M_{−}\} \end{aligned}

is a central ideal of {P, P }, which we term the full skew-dihedral homology group of P and denote by HF(P).

Proposition 5.11. Assume that ( , ) is an invariant form on a JKP P. Then there is a unique invariant symmetric bilinear form f on the Lie algebra {P, P } such that

(5.12)  \begin{matrix}f(\{a, b\}, \{c, d\}) = (δ(a, b)c, d), \\ f(\{a, b\}, \{x, y\}) = (δ(a, b)x, y) = (x, δ(a, b)y), \\ f(\{x, y\}, \{z, w\}) = (υ(x, y)z, w) \end{matrix}

for a, c ∈ J_{+}, b, d ∈ J_{−} x, z ∈ M_{+}, y, w ∈ M_{−}.

Proof. There is a unique form f̄ on the vector space (J_{+} ⊗J_{−}) ⨁(M_{+} ⊗M_{−}) such that

f̄(a ⊗ b, c ⊗ d) = (δ(a, b)c, d),
f̄(a ⊗ b, x ⊗ y) = (δ(a, b)x, y),
f̄(x ⊗ y, a ⊗ b) = (υ(x, y)a, b),
f̄(x ⊗ y, z ⊗ w) = (υ(x, y)z, w).

First, we show that f̄ is symmetric. Indeed for a, c ∈ J_{+}, b, d ∈ J_{−},

\begin{aligned}f̄(a ⊗ b, c ⊗ d) & = (δ(a, b)c, d) = (δ(c, b)a, d) = −(a, δ(c, b)d) = −(a, δ(c, d)b) \\ & = (δ(c, d)a, b) = f̄(c ⊗ d, a ⊗ b). \end{aligned}

Secondly, we note that for a ∈ J_{+}, b ∈ J_{−} x ∈ M_{+}, y ∈ M_{−}, (IF5) and (IF3) imply

\begin{aligned}f̄(a ⊗ b, x ⊗ y) & = (δ(a, b)x, y) = (a ∘ (b ∘ x), y) = (x, b ∘ (a ∘ y)) = (κ(a ∘ y, x), b) \\ & = (υ(x, y)a, b) = f̄(x ⊗ y, a ⊗ b). \end{aligned}

To complete the argument, we observe that for x, z ∈ M_{+}, y, w ∈ M_{−} it follows from (IF4) that

(x, κ(y, w) ∘ z) = (κ(z, x), κ(y, w)) = (w, κ(z, x) ∘ y).

This last identity, along with (JK3) and (IF2), implies that

f̄(x ⊗ y, z ⊗ w) = (υ(x, y)z, w)
= (υ(z, y)x, w) + (κ(x, z) ∘ y, w)
= (x, υ(y, z)w) − (x, κ(y, w) ∘ z)
= (x, υ(w, z)y) + (x, κ(y, w) ∘ z) − (x, κ(y, w) ∘ z)
= (υ(z, w)x, y) = f̄(z ⊗ w, x ⊗ y).

Now an easy verification shows that the subspace 𝒰 is contained in the radical of this form. Hence there is an induced form f on {P, P } having the properties in (5.12). To complete the proof it suffices to argue that 𝔦𝔫𝔰𝔱𝔯(P) is represented by skew-symmetric transformations of {P, P } (see Remark 5.7). Assume that T ∈ 𝔦𝔫𝔰𝔱𝔯(P), a, c ∈ J_{+}, b, d ∈ J_{−}. Then from (P6), (IF1-4), and Lemma 3.9 we have

f(T {a, b}, {c, d}) = f({T a, b} + {a, T b}, {c, d})
= f(δ(T a, b)c, d) + f(δ(a, T b)c, d)
= f(T δ(a, b)c, d) − f(δ(a, b)T c, d)
= −f(δ(a, b)c, T d) − f(δ(a, b)T c, d)
= −f({a, b}, {c, T d}) − f({a, b}, {T c, d})
= −f({a, b}, T {c, d})

as desired. For the other choices of elements from {P, P }, the calculation is virtually identical and consequently is omitted here.

The Lie algebra 𝔏̂(J, M)

5.13 Next we construct a Lie algebra 𝔏̂(J, M) using the algebra {P, P } instead of 𝔦𝔫𝔰𝔱𝔯(J, M). On the space

(5.14)  𝔏̂(J, M) = J_{−} ⊕ M_{−} ⊕ {P, P } ⊕ M_{+} ⊕ J_{+}

there is a unique anticommutative product [ , ] such that {P, P } is a subalgebra of 𝔏̂(J, M) and the following formulas hold:

(5.15) \begin{matrix}[p, (y, b)_{−}] = (λ(p)y, λ(p)b)_{−}, & [p, (x, a)_{+}] = (λ(p)x, λ(p)a)_{+}, \\ [(y, b)_{−}, (w, d)_{−}] = (0, κ(y, w))_{−}, & [(x, a)_{σ}, (z, c)_{σ}] = (0, κ(x, z))_{σ}, \\ [(x, a)_{+}, (y, b)_{−}] = (−b ∘ x, 0)_{−} & + \{a, b\} + \{x, y\} + (a ∘ y, 0)_{+} , \end{matrix}

for all pairs (x, a)_{+}, (z, c)_{+} ∈ M_{+} ⊕ J_{+}, (y, b)_{−}, (w, d)_{−} ∈ M_{−} ⊕ J_{−}, and all p ∈ {P, P }. Here λ is the homomorphism in Proposition 5.8.

Theorem 5.16. Let P = (J, M) be a Jordan-Kantor pair.

(i) The space 𝔏̂ = 𝔏̂(J, M) with the product in (5.15) is a Lie algebra. It has a 5-grading 𝔏̂ = 𝔏̂_{−2} ⊕ 𝔏̂_{−1} ⊕ 𝔏̂_{0} ⊕ 𝔏̂_{1} ⊕ 𝔏̂_{2} , where 𝔏̂_{−2} = J_{−}, 𝔏̂_{−1} = M_{−}, 𝔏̂_{0} = {P, P }, 𝔏̂_{1} = M_{+} , 𝔏̂_{2} = J_{+} .

(ii) Z(𝔏̂) ∩ 𝔏̂_{0} = HF(P) for the center Z(𝔏̂) of 𝔏̂. Moreover, if the algebra 𝔦𝔫𝔰𝔱𝔯(J, M) contains a grading derivation then Z(𝔏̂) = HF(P).

Proof. The fact that 𝔏̂ is a graded algebra follows directly from (5.15). We write |X| = i if X ∈ 𝔏̂_{i}. Define the linear map ρ : 𝔏̂ → 𝔏(J, M) by

(5.17)  ρ((y, b)_{−} + p + (x, a)_{+}) = (y, b)_{−} + λ(p) + (x, a)_{+} .

Evidently, ρ is an extension of λ and ker(ρ) ⊆ {P, P }. Moreover, it is easy to see using (4.2) and (5.15) that ρ is a homomorphism. It follows that 𝔧(X, Y, Z) = [[X, Y], Z] + [[Y, Z], X] + [[Z, X], Y] is zero whenever |X| + |Y | + |Z| ≠ 0.

Assume that |X| + |Y | + |Z| = 0. Because 𝔧(X, Y, Z) is an alternating function and because of the +/− symmetry, it suffices to check that 𝔧(X, Y, Z) = 0 in the following cases: (|X|, |Y |, |Z|) = (2, −1, −1) or (0, 2, −2) or (0, 1, −1). Here are the calculations in those particular cases:

𝔧((0, a)_{+}, (y, 0)_{−}, (w, 0)_{−}) = {a ∘ y, w} − {a, κ(y, w)} − {a ∘ w, y} = 0 by (P5),
𝔧(p, (0, a)_{+}, (0, b)_{−}) = {λ(p)a, b} − λ(p){a, b} + {a, λ(p)b} = 0 by (P6),
𝔧(p, (x, 0)_{+}, (y, 0)_{−}) = {λ(p)x, y} − λ(p){x, y} + {x, λ(p)y} = 0 by (P7).

To prove (ii) we note that the inclusion HF(P) = ker(λ) ⊆ Z(𝔏̂) ∩ 𝔏̂_{0} follows from (5.15). On the other hand, if z ∈ Z(𝔏̂) ∩ 𝔏̂_{0} , then for every i ≠ 0 and y_{i} ∈ 𝔏̂_{i} one has λ(z)y_{i} = [z, y_{i}] = 0. Thus, z ∈ HF(P).

Finally, if h = ∑_{i} δ(a_{i}, b_{i}) + ∑_{j} υ(x_{j}, y_{j}) ∈ 𝔦𝔫𝔰𝔱𝔯(J, M) is a grading derivation, then the element ĥ = ∑_{i}{a_{i}, b_{i}} + ∑_{j}{x_{j}, y_{j}} is a grading element of 𝔏̂(J, M). In particular, for every z ∈ Z(𝔏̂), one has [z, ĥ] = 0, hence z ∈ 𝔏̂_{0} . Therefore, Z(𝔏̂) = HF(P).

The algebra 𝔏̂(J, M) just constructed is universal in the following sense.

Theorem 5.18. Let L = ⨁_{i=−2}^{2} L_{i} be a 5-graded Lie algebra and (J, M) be a JKP. For any JKP homomorphism ρ : (J, M) → ((L_{−2}, L_{2}), (L_{−1}, L_{1})) there is a unique Lie algebra homomorphism from 𝔏̂ = 𝔏̂(J, M) to L which extends ρ. Proof. Let ρ(x) = x′ be such a JKP homomorphism. It follows from the assumptions that

\begin{matrix}\{a, b, c\}′ = [[a′, b′], c′], & \{x, y, z\}′ = [[x′, y′], z′], \\ (a ∘ x)′ = [a′, x′], & κ(x, z)′ = [x′, z′], \end{matrix}

for all a, c ∈ J_{σ} and b ∈ J_{−σ} and x, z ∈ M_{σ} and y ∈ M_{−σ}.

There is a unique linear map φ : (J_{+} ⊗ J_{−}) ⨁(M_{+} ⊗ M_{−}) → [L_{2}, L_{−2}] + [L_{1}, L_{−1}] such that φ(a ⊗ b) = [a′, b′] and φ(x ⊗ y) = [x′, y′]. To show that it induces a map ψ from {P, P } to L_{0} we need to prove that φ(𝒰) = 0 where 𝒰 is as in (5.2).

For elements of type (S1) we have

φ(δ(a, b)c ⊗ d + c ⊗ δ(a, b)d + δ(c, d)a ⊗ b + a ⊗ δ(c, d)b)

= [(δ(a, b)c)′, d′] + [c′, (δ(a, b)d)′] + [(δ(c, d)a)′, b′] + [a′, (δ(c, d)b)′] = [[[a′, b′], c′], d′] − [c′, [[b′, a′], d′]] + [[[c′, d′], a′], b′] − [a′, [[d′, c′], b′]] = 0.

The verification for elements of type (S2) is completely analogous.

For (S3), observe that

\begin{aligned}φ(κ(z, x) & ⊗ b − x ⊗ b ∘ z + z ⊗ b ∘ x) \\ & = [[z′, x′], b′] − [x′, [b′, z′]] + [z′, [b′, x′]] = 0 \end{aligned}

and similarly for (S4). Thus, φ(𝒰) = 0. Define a map ψ : 𝔏̂ → L to be ρ on 𝔏̂_{i}, for i ≠ 0, and ψ|_{{P,P }} to be the map induced by φ. We claim that ψ is a homomorphism.

First, we note that the formulas ψ([a, b]) = [ψ(a), ψ(b)] and ψ([x, y]) = [ψ(x), ψ(y)] follow immediately from the definition of ψ. Next, observe that

ψ([{a, b}, c]) = ψ(δ(a, b)c) = [[a′, b′], c′] = [ψ({a, b}), ψ(c)],

ψ([{a, b}, d]) = ψ(δ(a, b)d) = −[[b′, a′], d′] = [ψ({a, b}), ψ(d)],

ψ([{a, b}, x]) = ψ(δ(a, b)x) = [a′, [b′, x′]] = [[a′, b′], x′] = [ψ({a, b}), ψ(x)], ψ([{a, b}, y]) = ψ(δ(a, b)y) = −[b′, [a′, y′]] = [[a′, b′], y′] = [ψ({a, b}), ψ(y)],

and similarly

ψ([{x, y}, a]) = [ψ({x, y}), ψ(a)],
ψ([{x, y}, b]) = [ψ({x, y}), ψ(b)],
ψ([{x, y}, z]) = [ψ({x, y}), ψ(z)],
ψ([{a, b}, y]) = [ψ({x, y}), ψ(w)].

These equations imply in particular that (λ(p)X)′ = [ψ(p), X′] for any X ∈ 𝔏̂_{i}, i ≠ 0, and p ∈ {P, P }. Therefore,

ψ([p, {a, b}]) = ψ({λ(p)a, b} + {a, λ(p)b})

= [(λ(p)a)′, b′] + [a′, (λ(p)b)′] = [[ψ(p), a′], b′] + [a′, [ψ(p), b′]]

= [ψ(p), [a′, b′]] = [ψ(p), ψ({a, b})],

and ψ([p, a, b]) = [ψ(p), ψ({x, y})]. Thus, ψ is indeed the desired homomorphism. The uniqueness of such a homomorphism is apparent.

5.19 Before proceeding further we recall a few general facts about central extensions of Lie algebras (see [22] and [27]). A central extension of a Lie algebra L is a pair (L̃, π), where L̃ is a Lie algebra and π : L̃ → L is a surjective homomorphism whose kernel is contained in the center of L̃. A covering (L̂, π) of L is a central extension for which L̂ is perfect, i.e. L̂ = [L̂, L̂]. The covering (L̂, π) is said to be universal if for every central extension (L̃, τ) of L, there is a homomorphism ψ : L̂ → L̃ so that τ ∘ ψ = π. A covering (L̂, π) is universal if and only if L̂ is centrally closed, i.e., every central extension of L̂ splits [22, pp. 14-15]. We refer to a universal covering as the universal central extension, as it is unique up to isomorphism.

5.20 For any JKP (J, M), the Lie algebra 𝔏̂(J, M) with the homomorphism ρ defined in (5.17) is a central extension of 𝔏(J, M). Now, assume that 𝔦𝔫𝔰𝔱𝔯(J, M) contains a grading derivation h, and h = ∑_{i} δ(a_{i}, b_{i}) + ∑_{j} υ(x_{j}, y_{j}). Then the element ĥ = ∑_{i}{a_{i}, b_{i}} + ∑_{j}{x_{j}, y_{j}} is a grading element of 𝔏̂(J, M). In this case 𝔏̂(J, M) is perfect and hence is a covering of 𝔏(J, M).

Theorem 5.21. Assume that 𝔦𝔫𝔰𝔱𝔯(J, M) contains a grading derivation h. Then the Lie algebra 𝔏̂ = 𝔏̂(J, M) is centrally closed.

Proof. Suppose (L̃, π) is a central extension of 𝔏̂, and denote ker(π) by I. Set D = ad ĥ, and let D̃ = ad h̃ for a preimage h̃ of ĥ. Since D|_{𝔏_{i}} =

iId, and D(D^{2} − 1)(D^{2} − 4) = 0, we have D̃(D̃^{2} − 1)(D̃^{2} − 4)(L̃) ⊆ I. It follows that D̃^{2}(D̃^{2} − 1)(D̃^{2} − 4) = 0 because I is central. As the polynomials x^{2}, x − 1, x + 1, x − 2, x + 2 are relatively prime, there is a decomposition

(5.22)  L̃ = L̃_{x+2} ⊕ L̃_{x+1} ⊕ L̃_{x^{2}} ⊕ L̃_{x−1} ⊕ L̃_{x−2},

where L̃_{p(x)} is the null space of p(D̃).

Denote L̃_{x−i} by L_{i} for i = −2, −1, 1, 2. Then π(L_{i}) ⊆ 𝔏̂_{i}, and π(L̃_{x^{2}}) ⊆ 𝔏̂_{0} . Since I ⊆ L̃_{x^{2}} , we have I ⋂ L_{i} = (0) and therefore we can define a linear map ρ : 𝔏̂_{−2} ⊕ 𝔏̂_{−1} ⊕ 𝔏̂_{1} ⊕ 𝔏̂_{2} → L̃ such that π ∘ ρ = Id.

Next, we observe that [L_{i}, L_{j}] ⊆ L_{i+j} when i+j ≠ 0, and [[L_{i}, L_{j}], L_{k}] ⊆ L_{i+j+k} when i + j + k ≠ 0, since ad D̃ is a derivation and ad D̃ |_{L_{i}}= iId. It follows that L = ∑_{i=−2}^{2} L_{i}, where L_{0} = [L_{−2}, L_{2}] + [L_{−1}, L_{1}], is a subalgebra of L̃. Moreover, because L_{0} ⊆ L̃_{x^{2}} , the algebra L = ⨁_{i=−2}^{2} L_{i} is 5-graded.

We claim that ρ : (J, M) → ((L_{−2}, L_{2}), (L_{−1}, L_{1})) is a JKP homomorphism. First, we note that (J, M) = ((𝔏̂_{−2}, 𝔏̂_{2}), (𝔏̂_{−1}, 𝔏̂_{1})) as a JKP. Let i, j, k ≠ 0 and x ∈ 𝔏̂_{i}, y ∈ 𝔏̂_{j}, and z ∈ 𝔏̂_{k}. Assume that i + j ≠ 0. Since π is a Lie algebra homomorphism and π ∘ ρ = Id, ρ([x, y]) − [ρ(x), ρ(y)] ∈ I. On the other hand, ρ([x, y]) − [ρ(x), ρ(y)] ∈ L_{i+j} and L_{i+j} ⋂ I = (0). Thus, ρ([x, y]) = [ρ(x), ρ(y)]. Similarly, if i + j + k ≠ 0, then ρ([[x, y], z]) = [[ρ(x), ρ(y)], ρ(z)] proving that ρ is a JKP homomorphism.

Now according to Theorem 5.18, there is a Lie algebra homomorphism ψ : 𝔏̂ → L ⊆ L̃ which extends ρ, so π ∘ ψ|_{𝔏_{i}} = Id for i ≠ 0. It follows that

π ∘ ψ = Id, because 𝔏̂ is generated by subspaces 𝔏̂_{i}, i ≠ 0. Thus the extension π : L̃ → 𝔏̂ splits.

Corollary 5.23. The Lie algebra 𝔏̂ = 𝔏̂(J, M) with the homomorphism ρ from (5.17) is the universal central extension of 𝔏(J, M).

Description of the BC_{1} -graded Lie algebras of type D_{1}

Theorem 5.24. Let P = (J, M) be a JKP and assume 𝔦𝔫𝔰𝔱𝔯(P) contains a grading element. Then for every subspace ℋ ⊆ HF(P), the Lie algebra 𝔏(P, ℋ) = J_{−} ⊕ M_{−} ⊕ ({P, P }/ℋ) ⊕ M_{+} ⊕ J_{+} is BC_{1} -graded of type D_{1} . Every BC_{1} -graded Lie algebra of type D_{1} can be obtained in this way.

Proof. For a JKP P = (J, M) and a subspace ℋ ⊆ HF(P), the Lie algebra 𝔏(P, ℋ) has a grading element and a 5-grading with the required property inherited from 𝔏̂(P). As a consequence, 𝔏(P, ℋ) is BC_{1} -graded of type D_{1} .

Conversely, consider an arbitrary BC_{1} -graded Lie algebra L = ⨁_{i=−2}^{2} L_{i} of type D_{1} . According to the Theorem 4.9, L is a central extension of 𝔏(P) for a JKP P such that 𝔦𝔫𝔰𝔱𝔯(P) contains a grading derivation. Therefore, there exists a homomorphism ψ from the universal central extension 𝔏̂ = 𝔏̂(P) into L. Since ker(ρ) ⊆ 𝔏̂_{0} and L is generated by the subspaces L_{i} with i ≠ 0, the map ψ is surjective. Furthermore, ker(ψ) ⊆ ker(ρ) ⊆ Z(𝔏̂) = HF(P). Thus, L ≅ 𝔏̂(P)/ℋ = 𝔏(P, ℋ), where ℋ = ker(ψ) ⊆ HF(P).

6. J-ternary algebras and

the classification of BC_{1} -graded Lie algebras of type C_{1}

6.1 In this section we introduce a nonassociative object needed for the coordinatization of BC_{1} -graded Lie algebras of type C_{1} . Jordan-Kantor pairs with an invertible element in the Jordan pair can be used for this purpose. Alternatively, since JKPs of this kind can be described by another set of axioms, these Lie algebras may be coordinatized by J-ternary algebras. Following [8], we elect to use J-ternary algebras for coordinates. We establish connections between Jordan-Kantor pairs and J-ternary algebras, which enable us to apply the results of previous sections to classify the BC_{1} -graded Lie algebras of type C_{1} .

Jordan-Kantor pairs and J-ternary algebras

6.2 First, we recall background information about J-ternary algebras and related Lie algebra constructions. Our exposition here follows [2] and [8].

Let 𝔍 be a Jordan algebra with unit element 𝔢, and 𝔐 be a special unital 𝔍-module; that is, 𝔢 ∘ x = x for x ∈ 𝔐 and

(JT0)  (ab) ∘ x = \frac{1}{2}(a ∘ (b ∘ x) + b ∘ (a ∘ x)).

for all a, b ∈ 𝔍, x ∈ 𝔐. The space 𝔐 is said to be a J-ternary algebra if there is a skew bilinear map ⟨ , ⟩ : 𝔐×𝔐 → 𝔍, and a trilinear map ⟨ , , ⟩ : 𝔐×𝔐×𝔐 → 𝔐 satisfying:

(JT1) a ∘ ⟨v, w⟩ = \frac{1}{2}⟨av, w⟩ + \frac{1}{2}⟨v, aw⟩,

(JT2) a⟨u, v, w⟩ = ⟨au, v, w⟩ − ⟨u, av, w⟩ + ⟨u, v, aw⟩, (JT3) ⟨u, w⟩v = ⟨w, v, u⟩ − ⟨u, v, w⟩, (JT4) ⟨u, v⟩w = ⟨u, v, w⟩ − ⟨v, u, w⟩, (JT5) ⟨⟨u, v, w⟩, x⟩ + ⟨w, ⟨u, v, x⟩⟩ = ⟨u, ⟨w, x⟩v⟩, (JT6) ⟨u, v, ⟨w, x, y⟩⟩ = ⟨⟨u, v, w⟩, x, y⟩+⟨w, ⟨v, u, x⟩, y⟩+⟨w, x, ⟨u, v, y, ⟩⟩.

We denote a J-ternary algebra by (𝔍, 𝔐) or by _{𝔍}𝔐. The first is useful in drawing parallels with JKPs, while the second is handy as a shorthand in more complicated expressions.

Remark. In the definition above and in what follows we adopt the left operator version of J-ternary algebras not the right operator version used in [2]. The correspondence between the two different notions of J-ternary algebras is explained in [8, Remark 3.17]. This switch of side is necessary to make the results of this section consistent with those in previous sections. Also, the left operator notation was used in [8] for the other BC_{r} -graded algebras.

One may note also that the use of \frac{1}{2} in (JT0) is not consistent with (JK1). In previous sections on JKPs we have avoided introducing \frac{1}{2} because it makes the identities for JKP and Lie algebra constructions simpler, but omitting this factor departs from traditional Jordan theory. Since we use the definition of J-ternary algebras given in [8], which follows usual Jordan customs, we have to bear this inconsistency.

6.3 A 5-graded Lie algebra 𝒦(𝔍, 𝔐), the so-called Tits-Kantor-Koecher Lie algebra, can be associated with any J-ternary algebra (𝔍, 𝔐). Here we recall the construction of this algebra from [2] and [8].

Let ℰ = End_{F}(𝔍) ⊕ End_{F}(𝔐) with its canonical embedding into the algebra End_{F}(𝔍 ⊕ 𝔐). For a ∈ 𝔍, define the multiplication operator Ł_{a} ∈ ℰ by Ł_{a}(x, b) = (\frac{1}{2}a · x, a.b). Setting Ł_{a,b} = 2(Ł_{ab} + [Ł_{a}, Ł_{b}]) we note that Ł_{a,b}(x, c) = (a ∘ (b ∘ x) + b ∘ (a ∘ x), 2((ab)c + (cb)a) − (ca)b)). Also, for x, y ∈ 𝔐, suppose Ł_{x,y} ∈ ℰ is given by Ł_{x,y}(z, a) = (⟨x, y, z⟩, ⟨x, a · y⟩).

For A ∈ ℰ, assume A^{ε} = A − 2Ł_{A𝔢}, where 𝔢 is the unit of the Jordan algebra 𝔍. Let 𝔰𝔱𝔯(_{𝔍}𝔐) denote the set of elements A of ℰ which satisfy

[A + A^{ε}, Ł_{a}] = Ł_{Aa+A^{ε}a} and (6.4) [A, Ł_{x,y}] = Ł_{Ax,y} + Ł_{x,A^{ε}y} (6.5)

for all a ∈ 𝔍 and x, y ∈ M.

As was observed in [2], 𝔰𝔱𝔯(_{𝔍}𝔐) is a Lie subalgebra of ℰ, and A ∈ ℰ lies in 𝔰𝔱𝔯(_{𝔍}𝔐) if and only if A satisfies the relations

(6.6)  A(ab) = (Aa)b + a(Ab) − {a, A𝔢, b}_{𝔍},
(6.7)  A(a ∘ u) = (Aa) ∘ u + a ∘ (A^{ε}u),
(6.8)  A⟨u, v, w⟩ = ⟨Au, v, w⟩ + ⟨u, A^{ε}v, w⟩ + ⟨u, v, Aw⟩,
(6.9)  A⟨u, v⟩ = ⟨Au, v⟩ + ⟨u, Av⟩,

for all a, b ∈ 𝔍 and u, v, w ∈ M, where {a, c, b}_{𝔍} = (ac)b + (bc)a − (ab)c. We use { , , }_{𝔍} for this trilinear operation because we want to reserve { , , } for operations in JKPs.

Note that (6.6) is equivalent to the identity

(6.10)  A{a, c, b}_{𝔍} = {Aa, c, b}_{𝔍} + {a, A^{ε}c, b}_{𝔍} + {a, c, Ab}_{𝔍}.

6.11 Let 𝔇 be a Lie subalgebra of 𝔰𝔱𝔯(_{𝔍}𝔐) which contains all the operators Ł_{a} and Ł_{x,y}. Now suppose

𝒦 = 𝒦(𝔍, 𝔐, 𝔇) = 𝔍̃ ⊕ 𝔐̃ ⊕ 𝔇 ⊕ 𝔐 ⊕ 𝔍,

where 𝔍̃ = {ã | a ∈ 𝔍} and 𝔐̃ = {x̃ | x ∈ 𝔐} are second copies of 𝔍 and 𝔐. Then 𝒦 can be equipped with an anticommutative multiplication [ , ] for which 𝔇 is a subalgebra by defining

\begin{matrix}[T, (x, a)] = (T x, T a), & [T, (x, a) ] = (T^{ε}x, T^{ε}a) , \\ [(x, a), (y, b)] = (0, −⟨x, y⟩)), & [(x, a) , (y, b) ] = (0, ⟨x, y⟩) , \\ [(x, a), (y, b) ] = (−b · x, & 0) + Ł_{x,y} + Ł_{a,b} + (a · y, 0), \end{matrix}

for T ∈ 𝔇, x, y ∈ 𝔐, a, b ∈ 𝔍. Relative to this product 𝒦(𝔍, 𝔐, 𝔇) is a 5graded Lie algebra: 𝒦 = 𝒦_{−2} ⊕ 𝒦_{−1} ⊕ 𝒦_{0} ⊕ 𝒦_{1} ⊕ 𝒦_{2} with 𝒦_{−2} = 𝔍̃, 𝒦_{−1} = 𝔐̃, 𝒦_{0} = 𝔇, 𝒦_{1} = 𝔐, 𝒦_{2} = 𝔍. Furthermore, Z(𝒦) ⋂ 𝒦_{0} = (0). Thus by Theorem 4.5, the algebra 𝒦 = 𝒦(𝔍, 𝔐, 𝔇) can be described in terms of a JKP (J, M) whose structure is determined by the J-ternary algebra in the following way:

Theorem 6.12. Assume (𝔍, 𝔐) is a J-ternary algebra. Then

(i) J = (J_{−}, J_{+}) with J_{+} = 𝔍 = J_{−} and with the product {a_{σ}, b_{−σ}, c_{σ}} = 2[(ab)c + (bc)a − (ac)b]_{σ} is a Jordan pair;

(ii) M = (M_{−}, M_{+}) with M_{+} = 𝔐 = M_{−} is a special J -bimodule relative

to the action a ∘ x = ax ∈ M_{σ} for a ∈ J_{σ}, x ∈ M_{−σ}; (iii) the pair P = (J, M) with the skew maps κ : M_{σ} × M_{σ} → J_{σ}, κ(x, y) =

−σ⟨x, y⟩, and the triple product {x, y, z} = σ⟨x, y, z⟩ for x, z ∈ M_{σ} and y ∈ M_{−σ} forms a JKP.

For a J-ternary algebra 𝔐 we denote the JKP described in this theorem by 𝒫(𝔍, 𝔐). Note that 𝔦𝔫𝔰𝔱𝔯(𝒫(𝔍, 𝔐)) contains the grading derivation 2Ł_{𝔢}.

6.13 To describe Jordan-Kantor pairs of the form 𝒫(𝔍, 𝔐) we introduce

Definition 6.14. Let (J, M) be a JKP. An element a_{σ} ∈ J_{σ} is said to be J-invertible if the linear maps Q_{a_{σ}} : J_{−σ} → J_{σ}, Q_{a_{σ}}(b_{−σ}) = {a_{σ}, b_{−σ}, a_{σ}}, and Ł_{a_{σ}} : M_{−σ} → M_{σ}, Ł_{a_{σ}}(y_{−σ}) = a_{σ} ∘ y_{−σ}, are invertible.

Note that in this case, J_{−} ≅ J_{+} and M_{−} ≅ M_{+} as vector spaces. Our term for such an element comes from Jordan theory. The first condition of the definition is merely the condition for a_{σ} to be an invertible element of the Jordan pair J (see [28] for the definition and further details).

Proposition 6.15. A JKP P = (J, M) is isomorphic to 𝒫(𝔍, 𝔐) if and only if P possesses a J-invertible element.

Proof. First we note that the identity element 𝔢 ∈ J_{+} = 𝔍 is a J-invertible element of the JKP 𝒫(𝔍, 𝔐).

To establish the converse we consider a JKP P = (J, M) with J-invertible element a_{σ}. Since Q_{a_{σ}} is invertible, there is a unique element a_{−σ} ∈ J_{−σ} such that {a_{σ}, a_{−σ}, a_{σ}} = 2a_{σ}.

For every element c_{σ} ∈ J_{σ} one has c_{σ} = Q_{a_{σ}}(b_{−σ}) for some b_{−σ} ∈ J_{−σ}, and therefore {a_{σ}, a_{−σ}, c_{σ}} = {a_{σ}, a_{−σ}, Q_{a_{σ}}(b_{−σ})} = {a_{σ}, b_{−σ}, Q_{a_{σ}}(a_{−σ})} = 2Q_{a_{σ}}(b_{−σ}) = 2c_{σ}. Here, the second equality follows from a Jordan pair identity {a, b, Q_{a}(c)} = {a, c, Q_{a}(b)} ([28, Eqn. (1) of Sec. 2.1]).

In addition, since Ł_{a_{σ}} is invertible, corresponding to any x_{σ} ∈ M_{σ}, there is a y_{−σ} ∈ J_{−σ} so that x_{σ} = a_{σ}∘y_{−σ} and a_{σ}∘(a_{−σ}∘x_{σ}) = a_{σ}∘(a_{−σ}∘(a_{σ}∘y_{−σ})) = (1/2){a_{σ}, a_{−σ}, a_{σ}} ∘ y_{−σ} = a_{σ} ∘ y_{−σ} = x_{σ} (which follows from (JK1)).

In a similar way one can prove that {a_{−σ}, a_{σ}, b_{−σ}} = 2b_{−σ} and a_{−σ} ∘ (a_{σ} ∘ y_{−σ}) = y_{−σ} for every b_{−σ} ∈ J_{−σ} and y_{−σ} ∈ J_{−σ}. These equalities imply that the element δ(a_{σ}, a_{−σ}) is a grading derivation of P.

It follows then that elements a_{σ}, a_{−σ} and δ(a_{σ}, a_{−σ}) form an 𝔰𝔩_{2} - triple in the Lie algebra 𝔏(P) := 𝔏(J, M, 𝔦𝔫𝔰𝔱𝔯(J, M)). Moreover, this 𝔰𝔩_{2} - triple satisfies the requirements of Proposition 2.4, as can be seen from the representation theory of 𝔰𝔩_{2} .

Thus, the algebra 𝔏(P) is isomorphic to one constructed from a J-ternary algebra _{𝔍}𝔐.

Finally, one can check that the operations on P and _{𝔍}𝔐 are related in the way described in Theorem 6.11.

6.16 Next we want to show that the Lie algebra constructions for (𝔍, 𝔐) and 𝒫(𝔍, 𝔐) are essentially the same.

Proposition 6.17. The map ρ : 𝔰𝔱𝔯(𝒫(𝔍, 𝔐)) → 𝔰𝔱𝔯(_{𝔍}𝔐) defined by ρ(T) = T |_{𝔍⊕𝔐} is a Lie algebra isomorphism. Moreover, the Lie algebra 𝔏(J, M, 𝔇) is isomorphic to the Lie algebra 𝒦(𝔍, 𝔐, ρ(𝔇)).

Proof. Assume T ∈ 𝔰𝔱𝔯(𝒫(𝔍, 𝔐)) and T |_{𝔍⊕𝔐} = 0. Then (STR2) implies that T (b_{−}) = (1/2){𝔢_{+}, T (b_{−}), 𝔢_{+}} = 0 for 𝔢_{+} = 𝔢 ∈ J_{+} and every b_{−} ∈ J_{−}, and (STR2) implies that T (y_{−}) = 𝔢_{−} ∘ (𝔢_{+} ∘ T (y_{−})) = 𝔢_{−} ∘ T (𝔢_{+} ∘ y_{−}) = 0 for 𝔢_{−} = 𝔢 ∈ J_{−}, 𝔢_{+} = 𝔢 ∈ J_{+} and every y_{−} ∈ M_{−}. Injectivity of ρ follows.

Comparing (6.7-6.10) with (STR1-STR2), one can see that for A ∈ 𝔰𝔱𝔯(_{𝔍}𝔐) the map (A^{ε}, A) from End(J_{−} ⊕M_{−} ⊕M_{+} ⊕J_{+}) belongs to the algebra 𝔰𝔱𝔯(𝒫(𝔍, 𝔐)). Thus ρ is surjective. Then one can readily check that the map ρ̄: (x, a) + D + (y, b) ↦ (x, a) + ρ(D) + (y, b) is an isomorphism of 𝔏(J, M, 𝔇) and 𝒦(𝔍, 𝔐, ρ(𝔇)).

Full skew-dihedral homology of J-ternary algebras

6.18 We proceed to construct the universal central extension of the Lie algebra 𝒦(𝔍, 𝔐). This can be accomplished using the fact that (𝔍, 𝔐) can be represented as a JKP 𝒫 = 𝒫(𝔍, 𝔐) and then applying the construction of {𝒫, 𝒫} from 5.2. However, the Jordan pair in 𝒫 actually comes from a unital Jordan algebra. This enables us to simplify one of the defining identities for {𝒫, 𝒫} responsible for “the Jordan pair” part of the picture.

Theorem 6.19. Let _{𝔍}𝔐 be a J-ternary algebra and 𝒫 = 𝒫(𝔍, 𝔐) be the corresponding JKP. Then the subspace 𝒰 of (𝔍 ⊗ 𝔍) ⨁(𝔐 ⊗ 𝔐) described in 5.2 is spanned by the elements

(JTS1) a^{2} ⊗ a − 𝔢 ⊗ a^{3} ,

(JTS2) ⟨x, y, z⟩ ⊗ w + z ⊗ ⟨y, x, w⟩ + ⟨z, w, x⟩ ⊗ y + x ⊗ ⟨w, z, y⟩,

(JTS3) ⟨x, z⟩ ⊗ b − x ⊗ (b ∘ z) + z ⊗ (b ∘ x),

(JTS4) a ⊗ ⟨y, w⟩ − (a ∘ y) ⊗ w + (a ∘ w) ⊗ y.

Here a, b ∈ J and x, y, z, w ∈ M.

Proof. The expressions in (JTS2-JTS4) are merely those in (S2-S4) written in terms of the J-ternary operations.

It is left to show that elements of the form (JTS1) and (S1) span the same subspace in (𝔍 ⊗ 𝔍) ⨁(𝔐 ⊗ 𝔐). Note that (S1) is simply a linearization of {a, b, a} ⊗ b − a ⊗ {b, a, b}. Using the bilinear product in 𝔍 instead of the trilinear one, we obtain the following form of this element,

(S1′) 2(ab)a ⊗ b − a^{2}b ⊗ b − 2a ⊗ (ba)b + a ⊗ b^{2}a

and hence (S1) and (S1′) span the same subspace, which we denote by 𝒰_{1} .

First, we prove that elements in (JTS1) belong to the span of those in (S1′). Setting b = c = d = 𝔢 in (S1′), we see that the elements

(6.20)  a ⊗ 𝔢 − 𝔢 ⊗ a

belong to 𝒰_{1} . Next, (S1′) with c = 𝔢 and b = d = a implies that

(6.21)  2a^{2} ⊗ a − a ⊗ a^{2} − 𝔢 ⊗ a^{3}

is in 𝒰_{1} ; and (S1′) with b = 𝔢 and c = d = a gives that

(6.22)  a^{2} ⊗ a − 2a ⊗ a^{2} + a^{3} ⊗ 𝔢

belongs to 𝒰_{1} . Subtracting (6.22) from (6.21) multiplied by 2 and using (6.20), we have that elements of the form (JTS1) are in 𝒰_{1} .

Now we prove that the elements in (S1′) belong to the subspace 𝒰_{2} spanned by elements (JTS1). For this we write X ≡ Y if X − Y is in 𝒰_{2} . To begin note that linearization of (JTS1) yields

(6.23)  ab ⊗ c + bc ⊗ a + ca ⊗ b ≡ 𝔢 ⊗ (ab)c + 𝔢 ⊗ (bc)a + 𝔢 ⊗ (ca)b.

This identity with b = c = 𝔢 gives us

(6.24)  a ⊗ 𝔢 ≡ 𝔢 ⊗ a,

while with b = a, c = 𝔢 it says

(6.25)  a ⊗ a ≡ 𝔢 ⊗ a^{2}.

Linearization of (6.25) provides us with the relation

(6.26)  a ⊗ b + b ⊗ a ≡ 2𝔢 ⊗ ab.

Now (6.25) and (6.24) imply that a ⊗ a^{2} − a^{3} ⊗ 𝔢 ≡ −a^{2} ⊗ a + 2𝔢 ⊗ a^{3} − a^{3} ⊗ 𝔢 ≡ −a^{2} ⊗ a + 𝔢 ⊗ a^{3} ≡ 0. It follows then that the subspace 𝒰_{2} is invariant under the linear transformation τ of (𝔍 ⊗ 𝔍) ⨁(𝔐 ⊗ 𝔐) which sends a ⊗ b to b ⊗ a and x ⊗ y to y ⊗ x.

We are now ready to prove that (S1′) is in 𝒰_{2} . Letting c = ab in (6.23) and applying (6.25) and (6.26), we have

(6.27)  (ab)a ⊗ b + a ⊗ (ba)b ≡ 𝔢 ⊗ ((ab)a)b − 𝔢 ⊗ ((ba)b)a.

On the other hand, (6.23) with a = a^{2} and c = b gives

(6.28)  2a^{2}b ⊗ b + b^{2} ⊗ a^{2} ≡ 2𝔢 ⊗ (a^{2}b)b + 𝔢 ⊗ b^{2}a^{2}.

Interchanging a and b in (6.28) and applying τ shows that

(6.29)  2a ⊗ b^{2}a + b^{2} ⊗ a^{2} ≡ 2𝔢 ⊗ (b^{2}a)a + 𝔢 ⊗ b^{2}a^{2}.

Then (6.29) subtracted from (6.28) gives

(6.30)  a^{2}b ⊗ b − a ⊗ b^{2}a ≡ 𝔢 ⊗ (a^{2}b)b − 𝔢 ⊗ (b^{2}a)a,

and subtracting (6.30) from (6.27) multiplied by 2 yields 2(ab)a ⊗ b − a^{2}b ⊗ b − 2a⊗(ba)b+a⊗b^{2}a ≡ 𝔢⊗(2((ab)a)b−(a^{2}b)b−2((ba)b)a+(b^{2}a)a). However, the

identity 2((ab)a)b − (a^{2}b)b − 2((ba)b)a + (b^{2}a)a = 0 being a partial linearization of a(ba^{2}) = (ab)a^{2} holds in any Jordan algebra 𝔍.

6.31. Let us denote (𝔍⊗𝔍⊕𝔐⊗𝔐)/𝒰 by {_{𝔍}𝔐,_{𝔍}𝔐} and set {a, b} = a⊗b+𝒰 and {x, y} = x ⊗ y + 𝒰.

We have a Lie algebra homomorphism ρ : {_{𝔍}𝔐,_{𝔍}𝔐} → 𝔰𝔱𝔯(_{𝔍}𝔐) defined by ρ({a, b}) = Ł_{a,b} and ρ({x, y}) = Ł_{x,y}. Naturally, we call the kernel of this homomorphism the full skew-dihedral homology group HF(𝔍, 𝔐) of the 𝔍-ternary algebra _{𝔍}𝔐.

The following result is a consequence of Proposition 6.17, Theorem 5.16, and Corollary 5.23.

Theorem 6.32. For a J-ternary algebra _{𝔍}𝔐, let N = {(x, a) | x ∈ 𝔐, a ∈ 𝔍} and suppose N is a second copy of N.

(i) The space 𝒦̂(𝔍, 𝔐) = N ⊕ {_{𝔍}𝔐,_{𝔍}𝔐} ⊕ N with the anticommutative product [ , ] defined by the formulas

(6.33)

[p, q] is the product in {_{𝔍}𝔐,_{𝔍}𝔐},
[q, (x, a)] = (ρ(q)(x), ρ(q)(a)),
[q, (x, a) ] = (ρ(q)^{ε}(x), ρ(q)^{ε}(a)) ,

[(x, a), (y, b)] = (0, −⟨x, y⟩) , [(x, a) , (y, b) ] = (0, ⟨x, y⟩) ,

[(x, a), (y, b) ] = −(b · x, 0) + {x, y} + {a, b} + (a · y, 0),

for p, q ∈ {_{𝔍}𝔐,_{𝔍}𝔐}, x, y ∈ 𝔐, a, b ∈ 𝔍, is a 5-graded Lie algebra. (ii) HF(𝔍, 𝔐) is the center of 𝒦̂(𝔍, 𝔐). (iii) 𝒦̂(𝔍, 𝔐) is the universal central extension of 𝒦(𝔍, 𝔐).

A description of the BC_{1} -graded algebras of type C_{1}

Theorem 6.34. Let _{𝔍}𝔐 be a J-ternary algebra and ℋ be a subspace of HF(𝔍, 𝔐). Then the Lie algebra 𝒦(𝔍, 𝔐, ℋ) = N ⊕ ({_{𝔍}𝔐,_{𝔍}𝔐}/ℋ) ⊕ N is BC_{1} -graded Lie algebra of type C_{1} , and every BC_{1} -graded algebra of type C_{1} can be obtained in this way.

Proof. The Lie algebra 𝒦 = 𝒦(𝔍, 𝔐, ℋ) is 5-graded with 𝒦_{0} = [𝒦_{−1}, 𝒦_{1}] + [𝒦_{−2}𝒦_{2}]. Consider the elements e = (0, 𝔢), f = (0, 2𝔢) , and h = 2{𝔢, 𝔢} + ℋ. It follows from (6.33) that ⟨e, f, h⟩ is an 𝔰𝔩_{2} -triple and h is a grading element. According to Theorem 2.9, 𝒦 is a BC_{1} -graded algebra of type C_{1} .

Conversely, assume L is a BC_{1} -graded algebra of type C_{1} with a grading subalgebra 𝔤 spanned by an 𝔰𝔩_{2} -triple ⟨e, f, h⟩. Since L is also BC_{1} -graded of type D_{1} by Theorem 2.9, L ≅ 𝔏(P, ℋ) for some JKP P = (J, M) and a subspace ℋ ⊆ HF(P). The image of e is an invertible element of J. Hence P ≅ 𝒫(𝔍, 𝔐) and L ≅ 𝔏(P, ℋ) ≅ 𝒦(𝔍, 𝔐, ℋ) for some J-ternary algebra _{𝔍}𝔐.

7. Kantor pairs, structurable algebras, and

the classification of BC_{1} -graded Lie algebras of type B_{1}

7.1 In this section we establish connections between Jordan-Kantor pairs and structurable algebras. This enables us to apply the results of previous sections to classify the BC_{1} -graded Lie algebras of type B_{1} .

Kantor pairs and Jordan-Kantor pairs

Definition 7.2. (See [9].) A pair of spaces M = (M_{−}, M_{+}) is said to be a Kantor pair if it possesses a triple product {x, y, z} = V_{x,y}z ∈ M_{σ} for x, z ∈ M_{σ} and y ∈ M_{−σ} such that

(KP1) [V_{x,y}, V_{z,w}] = V_{V_{x,y}z,w} − V_{z,V_{y,x}w}, (KP2) K_{x,z}V_{w,u} + V_{u,w}K_{x,z} = K_{K_{x,z}w,u} hold for K_{x,z}y := {x, y, z} − {z, y, x} and u ∈ M_{σ}, w ∈ M_{−σ}.

7.3 We briefly describe the various relationships among the notions of Jordan, Kantor, and Jordan-Kantor pairs.

The triple product V_{a,b}c = {a, b, c} in a Jordan pair J = (J_{−}, J_{+}) satisfies (KP1), so a Jordan pair is just a special case of a Kantor pair with K_{x,z} = 0 for all x, z.

Assume now that (J, M) is a JKP and let K_{x,z}y = κ(x, z) ∘ y =^{(JK3)} {x, y, z} − {z, y, x}. Identity (JK2) is the same as (KP1), and identity (JK5) implies (KP2). Therefore, the pair M = (M_{−}, M_{+}) with its product V_{x,y}z = {x, y, z} and with K_{x,z}y = κ(x, z) ∘ y is a Kantor pair. Thus, any JKP (J, M) consists of a Jordan pair J and a Kantor pair M whose structures are interrelated via ∘ and κ and the identities listed in (3.1).

7.4 Any Kantor pair M = (M_{−}, M_{+}) can be embedded into a JKP 𝒫(M) in the following way. The exposition below is sketchy because we will not need it later on.

The 5-graded Lie algebra 𝔏 = 𝔏_{−2} ⊕ 𝔏_{−1} ⊕ 𝔏_{0} ⊕ 𝔏_{1} ⊕ 𝔏_{2} constructed from M in [9, Sec. 4] satisfies all the conditions of Theorem 4.5 (iv). Thus, it determines a JKP 𝒫(𝔏) = ((𝔏_{−2}, 𝔏_{2}), (𝔏_{−1}, 𝔏_{1})). It follows from the construction that the Kantor pair (𝔏_{−1}, 𝔏_{1}) is isomorphic to M and that (𝔏_{−2}, 𝔏_{2}) is the Jordan pair K = (K_{−}, K_{+}), where K_{σ} is the subspace of Hom(M_{−σ}, M_{σ}) spanned by the operators K_{x,z} with the product {A, B, C} = ABC + CBA. The action of K on M is given K_{x,z} ∘ y = K_{x,z}(y). The skew maps κ : M_{σ} × M_{σ} → K_{σ} are defined by κ(x, y) = K_{x,y}. We denote the JKP obtained in this manner by 𝒫(M). These JKPs admit the following description.

Proposition 7.5. A JKP (J, M) is isomorphic to 𝒫(M) for some Kantor pair M if and only if

(i) J acts faithfully on the bimodule M, and (ii) J_{σ} = κ(M_{σ}, M_{σ}) for σ = ±.

Proof. The “only if” part follows from the description of 𝒫(M). For the other direction note that if (i) and (ii) hold, then a straightforward verification shows that the map ρ : (J, M) → 𝒫(M) defined by

ρ((j_{−}, m_{−}, m_{+}, j_{+})) = (Ł_{j_{−}}, m_{−}, m_{+}, Ł_{j_{+}}),

where Ł_{j_{σ}}(m_{−σ}) = j_{σ} ∘ m_{−σ}, is an isomorphism.

Structurable algebras

7.6. Structurable algebras can be considered as a subclass of Kantor pairs, and as such, as a subclass of JKPs. They play an important role in our classification of BC_{1} -graded algebras of type B_{1} as we have already seen in Theorem 2.6.

Recall that a unital algebra (X,^{−}) with an involution “^{−} ” is said to be structurable provided the operators V_{x,y}z = {x, y, z} = (xȳ)z + (zȳ)x − (zx̄)y satisfy identity (KP1).

If (X,^{−}) is a structurable algebra, then according to [9], the pair of spaces (X, X) with the triple product { , , } forms a Kantor pair. As in (7.4), we can associate a JKP to (X,^{−}). This time we provide more details.

7.7. We start with a description of the corresponding Lie algebra construction (see [3] for details). For a structurable algebra (X,^{−}), the set S = {s ∈ X | s̄ = −s} of skew-hermitian elements is a Jordan triple system with respect to the product {r, s, t} = (rs)t + (ts)r. That is to say, the pair of spaces (S, S) with this product is a Jordan pair.

It follows from (KP1) that the subspace V_{X,X} of End(X) spanned by operators V_{x,y} is a Lie subalgebra of End(X). Let N = {(x, s) | x ∈ X, s ∈ S}, and denote a second copy of N by N . On the vector space

𝒦 = N ⊕ V_{X,X} ⊕ N

define the anti-commutative bilinear product [ , ] which extends the product on V_{X,X} and satisfies

(7.8)  \begin{matrix}[V_{x,y}, (z, s)] = (V_{x,y}(z), x(ȳs) + (sy)x̄), \\ [V_{x,y}, (z, s) ] = (−V_{y,x}(z), −y(x̄s) − (sx)ȳ) , \\ [(x, r), (y, s)] = (0, xȳ − yx̄) ,  [(x, r) , (y, s) ] = (0, xȳ − yx̄) , \\ [(x, r), (y, s) ] = −(sx, 0) + V_{x,y} + Ł_{r}Ł_{s} + (ry, 0), \end{matrix}

for x, y, z ∈ X, r, s ∈ S, where Ł_{s} denotes left multiplication by s and Ł_{r}Ł_{s} = (1/2)(V_{rs,1} − V_{r,s}). Note: The construction above is from [3]. It differs from [8] by some multiples of 2.

The Lie algebra defined above is referred to as the Tits-Kantor-Koecher Lie algebra of (X,^{−}), and usually it is denoted 𝒦(X,^{−}) or 𝒦 for short. It is not difficult to see that it possesses a 5-grading

𝒦 = 𝒦_{−2} ⊕ 𝒦_{−1} ⊕ 𝒦_{0} ⊕ 𝒦_{1} ⊕ 𝒦_{2},

where 𝒦_{−2} = (0, S) , 𝒦_{−1} = (X, 0) , 𝒦_{0} = V_{X,X}, 𝒦_{1} = (X, 0), 𝒦_{2} = (0, S).

7.9. Let us calculate the JKP 𝒫(𝒦). From the formulas in (7.8) we can read off the operations in 𝒫(𝒦). The Jordan pair (𝒦_{−2}, 𝒦_{2}) is simply the Jordan triple system S with the product {r, s, t} = (rs)t + (ts)r, because

[[(0, r), (0, s) ], (0, t)] = \frac{1}{2}[V_{rs,1} − V_{r,s}, (0, t)]

= \frac{1}{2}(0, t(sr) + (rs)t + (ts)r + r(st))

= (0, (rs)t + (ts)r).

The last equality follows from the identity t(sr) + r(st) = (rs)t + (ts)r which is valid in any structurable algebra. Next, the Kantor pair (𝒦_{−1}, 𝒦_{1}) is just the Kantor triple system X under the product {x, y, z} = V_{x,y}z. The action of S on X is given by s ∘ x = sx and the map κ is defined by κ(x, z) = xz̄− zx̄. Let us denote this JKP by 𝒫(X,^{−}). Thus, 𝒫(X,^{−}) = 𝒫(𝒦(X,^{−})). In addition V_{1,1} is a grading derivation in 𝔦𝔫𝔰𝔱𝔯(𝒫(X,^{−})). To describe all JKPs obtained from structurable algebras we require the following concept.

Definition 7.10. Let (J, M) be a JKP. An element x_{σ} ∈ M_{σ} is said to be conjugate invertible provided συ(x_{σ}, x_{−σ}) is a grading derivation of (J, M) for some x_{−σ} ∈ M_{−σ}.

Note that the element 1 ∈ X is conjugate invertible in the JKP 𝒫(X,^{−}) because ad υ(1, 1) |_{𝒦_{i}} = i Id |_{𝒦_{i}} .

Proposition 7.11. A JKP P = (J, M) is isomorphic to 𝒫(X,^{−}) for a structurable algebra (X,^{−}) if and only if there is a conjugate invertible element x_{σ} ∈ M_{σ}. In this case 𝔏(P) ≅ 𝒦(X,^{−}). Proof. We need to prove only the “if” direction. Let 𝔏 = 𝔏(P) be the Lie algebra constructed from a JKP P = (J, M), and assume P contains a conjugate invertible element x_{σ} ∈ M_{σ}. It follows from (4.2) that

e = (2x_{σ}, 0), f = (x_{−σ}, 0), h = 2συ(x_{σ}, x_{−σ})

is an 𝔰𝔩_{2} -triple in 𝔏.

One can verify that this 𝔰𝔩_{2} -triple satisfies the conditions of Proposition 2.5 and therefore P ≅ 𝒫(𝔏(P)) ≅ 𝒫(𝒦(X,^{−})) = 𝒫(X,^{−}).

As we have just shown, structurable algebras provide particular examples of JKPs. In the rest of this section we see how the general considerations of Sections 4-7 greatly simplify in this special case. We begin with an easier construction of the structure algebra. Assume that (X,^{−}) is a structurable algebra. It follows from the isomorphism in Proposition 7.11 that 𝔦𝔫𝔰𝔱𝔯(𝒫(X,^{−})) ≅ V_{X,X}. Now we want to realize this isomorphism explicitly. In fact we will establish that 𝔰𝔱𝔯(𝒫(X,^{−})) is isomorphic to the structure Lie algebra of (X,^{−}) which is described in [3, Cor. 5] as follows:

𝔰𝔱𝔯(X,^{−}) = {E ∈ End(X) | [E, V_{x,y}] = V_{Ex,y} + V_{x,E′y} for some E′ ∈ End(X)}.

Proposition 7.12. Let (X,^{−}) be a structurable algebra. The map ϕ : T ↦ T |_{X} is an isomorphism from 𝔰𝔱𝔯(𝒫(X,^{−})) onto 𝔰𝔱𝔯(X,^{−}). Moreover, ϕ(𝔦𝔫𝔰𝔱𝔯(𝒫(X,^{−}))) = V_{X,X}. Proof. We use the auxiliary fact that if y ∈ X has the property that {x, y, z} = 0 for all x, z ∈ X then y = 0. Indeed letting x = z = 1, we have {1, y, 1} = 2ȳ − y = 0. Applying the involution to 2ȳ = y we see that 2y = ȳ and hence 2y = ȳ = (1/2)y. Thus, y = 0.

Let us show that the map ϕ : T ↦ T |_{X} from 𝔰𝔱𝔯(𝒫(X,^{−})) to End(X) is injective. In this calculation we identify (X, 0), (X, 0) , (0, S), and (0, S) with X, X , S, and S . It follows from (STR1) that {x, T (y ), z} = T {x, y , z} − {T x, y , z} − {x, y , T z}. If T |_{X} = 0, then {x, T (y ), z} = 0 and therefore T (y ) = 0 for every y . For this T, the equations T |_{S} = 0 and T |_{S} = 0 follow from (STR2). Hence T = 0. Surjectivity of ϕ is clear. Thus ϕ is an isomorphism.

To prove the second assertion we note that V_{X,X} ⊆ ϕ(𝔦𝔫𝔰𝔱𝔯(𝒫(X,^{−}))). On the other hand, for every r, s ∈ S we claim that

(7.13)  2δ(r, s) = υ(rs, 1) − υ(r, s).

According to the discussion above, it is necessary to check (7.13) just on X. However, on X it amounts to 2Ł_{r}Ł_{s} = V_{rs,1} − V_{r,s} which is established in [3, Eq. 14].

Full skew-dihedral homology of structurable algebras

The universal central extension of the Lie algebra 𝒦(X,^{−}) is described in [10, Cor. 5.19]. Our aim here is to give an alternative description which is a consequence of the general theory developed for JKPs in Section 5.

We observe how the definition of the Lie algebra {𝒫, 𝒫} simplifies for a structurable algebra viewed as a JKP.

Proposition 7.14. If (X,^{−}) is a structurable algebra and 𝒫 = 𝒫(X,^{−}), then {𝒫, 𝒫} ≅ (X ⊗ X)/ℛ where ℛ is the subspace of X ⊗ X spanned by the elements of the form

(R1) V_{x,y}z ⊗ w − z ⊗ V_{y,x}w + V_{z,w}x ⊗ y − x ⊗ V_{w,z}y

for x, y, z, w ∈ X

Proof. Recall that {𝒫, 𝒫} = ((S ⊗ S) ⨁(X ⊗ X))/𝒰 where 𝒰 is the subspace spanned by the elements of the form (S1-S4) in (5.2). In this case S is a subspace of X, and therefore to avoid confusion we need to distinguish elements from X ⊗ X and S ⊗ S. We use the subscripts X and S for this purpose; thus (r ⊗ s)_{X} ∈ X ⊗ X but (r ⊗ s)_{S} ∈ S ⊗ S.

First we simplify the set of generators for 𝒰 by showing that 𝒰 is spanned by the elements of the form (R1) and

(R2) 2(r ⊗ s)_{S} − (rs ⊗ 1)_{X} + (r ⊗ s)_{X}

for r, s ∈ S.

It is easy to see that the subspace 𝒰_{1} spanned by the elements in (R1-R2) is contained in 𝒰. Indeed, (R1) and (S2) are the same, and if we set w = 1, y = s, a = r in (S4) we obtain (R2).

To establish the reverse containment, it suffices to verify that the generators (S1-S4) of 𝒰 belong to 𝒰_{1} . In this proof we write ξ ≡ ζ to mean ξ −ζ ∈ 𝒰_{1} .

In particular, from (5.3) we see that (R1) and (R2) are equivalent to the following in this notation:

(7.15)  υ(x, y)(z ⊗ w)_{X} ≡ −υ(z, w)(x ⊗ y)_{X} ;

(7.16)  2(r ⊗ s)_{S} ≡ (rs ⊗ 1)_{X} − (r ⊗ s)_{X}.

Using these relations and (7.13) we obtain that

υ(x, y)(r ⊗ s)_{S}+δ(r, s)(x ⊗ y)_{X}

≡ \frac{1}{2}υ(x, y)(rs ⊗ 1)_{X} − \frac{1}{2}υ(x, y)(r ⊗ s)_{X} + δ(r, s)(x ⊗ y)_{X}

≡ −\frac{1}{2}υ(rs, 1)(x ⊗ y)_{X} + \frac{1}{2}υ(r, s)(x ⊗ y)_{X} + δ(r, s)(x ⊗ y)_{X}

≡ 0.

In turn, this identity (along with (7.13) and (7.16)) implies that

δ(r, s)(t ⊗ u)_{S} + δ(t, u)(r ⊗ s)_{S}

≡ \frac{1}{2}δ(r, s)(tu ⊗ 1)_{X} − \frac{1}{2}δ(r, s)(t ⊗ u)_{X} + δ(t, u)(r ⊗ s)_{S}

≡ −\frac{1}{2}υ(tu, 1)(r ⊗ s)_{S} + \frac{1}{2}υ(t, u)(r ⊗ s)_{S} + δ(t, u)(r ⊗ s)_{S}

≡ 0.

Thus, elements of the form (S1) belong to 𝒰_{1} . As (S2) is the same as (R1), those elements belong to 𝒰_{1} also.

Next we note that (7.15) with z = w = 1 gives us

υ(x, y)(1 ⊗ 1)_{X} ≡ −2(x ⊗ y)_{X} .

We use this identity and relations (7.16) and (7.13) to modify (S3) as follows:

(κ(z, x) ⊗ s)_{S} − (x ⊗ sz)_{X} + (z ⊗ sx)_{X}

≡ \frac{1}{2}(κ(z, x)s ⊗ 1)_{X} − \frac{1}{2}(κ(z, x) ⊗ s)_{X} − (x ⊗ sz)_{X} + (z ⊗ sx)_{X}

≡ −\frac{1}{2}(\frac{1}{2}υ(κ(z, x)s, 1) − \frac{1}{2}υ(κ(z, x), s) − υ(x, sz) + υ(z, sx))(1 ⊗ 1)_{X}

≡ −\frac{1}{2}(δ(κ(z, x), s) − υ(x, sz) + υ(z, sx))(1 ⊗ 1)_{X}

It is left only to note that δ(κ(z, x), s) − υ(x, sz) + υ(z, sx) = 0, because (δ(κ(z, x), s) − υ(x, sz) + υ(z, sx))|_{X} = Ł_{κ(z,x)}Ł_{s} − V_{x,sz} + V_{z,sx} = 0 in any structurable algebra (see [4, Eq. (14)]). Thus, (S3) is in 𝒰_{1} .

The proof for (S4) is similar with the sole exception being that one applies the identity Ł_{s}Ł_{κ(z,x)} − V_{sx,z} + V_{sz,x} = 0, which follows from [4, Eqs. (4), (8), and (14)].

Now we claim that ℛ = (X ⊗ X) ∩ 𝒰. The inclusion ℛ ⊆ (X ⊗ X) ∩ 𝒰 is clear. To show the reverse inclusion it suffices to prove that ∑_{i} (r_{i} ⊗ s_{i})_{S} = 0 implies that ∑_{i} (r_{i}s_{i} ⊗ 1)_{X} − ∑_{i} (r_{i} ⊗ s_{i})_{X} = 0. Here we can assume that {s_{i}} is linearly independent, and therefore every r_{i} = 0, which proves our claim.

It follows then that the kernel of the map ψ : X ⊗ X → {𝒫, 𝒫} = ((S ⊗ S) ⊕ (X ⊗ X))/𝒰 defined by x ⊗ y ↦ x ⊗ y + 𝒰 is precisely ℛ. Surjectivity of ψ follows from (7.16).

7.17. Let us denote (X ⊗ X)/ℛ by {X, X} and {x, y} = x ⊗ y + ℛ. We have a chain of homomorphisms: ψ : {X, X} → {𝒫, 𝒫} defined in 7.14, λ : {𝒫, 𝒫} → 𝔰𝔱𝔯(𝒫(X,^{−})) defined in 5.8, and ϕ : 𝔰𝔱𝔯(𝒫(X,^{−})) → 𝔰𝔱𝔯(X,^{−}) defined in 7.12. We denote their composition by ϑ. Thus ϑ({x, y}) = V_{x,y}. Define the full skew-dihedral homology group HF(X,^{−}) of a structurable algebra (X,^{−}) to be the kernel of the map ϑ; that is,

HF(X,^{−}) = {∑_{i}{x_{i}, y_{i}} ∈ {X, X} | ∑_{i} V_{x_{i},y_{i}} = 0}.

This definition agrees with the original definition from [10] because HF(X,^{−}) is the center of the universal central extension of 𝒦(X,^{−}) (see Theorem 7.18 below).

The following result is a consequence of Theorem 5.16, Propositions 7.11 and 7.14, and Corollary 5.23.

Theorem 7.18. For a structurable algebra (X,^{−}) with skew-hermitian elements S, let N = {(x, s) | x ∈ X, s ∈ S} and suppose N is a second copy of N.

(i) The space 𝒦̂(X,^{−}) = N ⊕{X, X}⊕N with the anticommutative product [ , ] defined by the formulas,

(7.19)  \begin{matrix}[\{x, y\}, \{z, w\}] = \{V_{x,y}z, w\} − \{z, V_{y,x}w\}, \\ [\{x, y\}, (z, s)] = (V_{x,y}(z), (sy)x̄ + x(ȳs), \\ [\{x, y\}, (z, s) ] = (−V_{y,x}(z), −(sx)ȳ − y(x̄s)) , \\ [(x, r), (y, s)] = (0, xȳ − yx̄) , \\ [(x, r) , (y, s) ] = (0, xȳ − yx̄) , \\ [(x, r), (y, s) ] = −(sx, 0) + \{x, y\} + (1/2)\{rs, 1\} \\ − (1/2)\{r, s\} + (ry, 0), \end{matrix}

is a 5-graded Lie algebra. (ii) HF(X,^{−}) is the center of 𝒦̂(X,^{−}).

(iii) 𝒦̂(X,^{−}) is the universal central extension of 𝒦(X,^{−}).

A description of the BC_{1} -graded algebras of type B_{1}

Theorem 7.20. Let (X,^{−}) be a structurable algebra and ℋ be a subspace of HF(X,^{−}). The Lie algebra 𝒦(X,^{−}, ℋ) = N ⊕({X, X}/ℋ)⊕N is BC_{1} -graded Lie algebra of type B_{1} , and every BC_{1} -graded algebra of type B_{1} can be obtained in this way. Proof. The Lie algebra 𝒦 = 𝒦(X,^{−}, ℋ) is 5-graded with 𝒦_{0} = [𝒦_{−1}, 𝒦_{1}] + [𝒦_{−2}𝒦_{2}]. It follows from (7.19) that the elements e = (2, 0), f = (1, 0) , and h = 2{1, 1} + ℋ determine an 𝔰𝔩_{2} -triple, and (1/2)h is a grading element. According to Theorem 2.9, 𝒦 is a BC_{1} -graded algebra of type B_{1} .

Conversely, assume L is a BC_{1} -graded algebra of type B_{1} with a grading subalgebra 𝔤 spanned by an 𝔰𝔩_{2} -triple ⟨e, f, h⟩. Since L is also of type D_{1} , L ≅ 𝔏(P, ℋ) for some JKP P = (J, M) and a subspace ℋ ⊆ HF(P). The image of e is a conjugate invertible element of M_{+} . Hence P ≅ 𝒫(X,^{−}) and L ≅ 𝔏(P, ℋ) ≅ 𝒦(X,^{−}, ϑ(ℋ)) for some structurable algebra (X,^{−}).

8. Derivations of BC_{1} -graded Lie algebras

8.1 In this section we describe the derivations of BC_{1} -graded Lie algebras. First, we consider a 5-graded Lie algebra L = ⨁_{i=−2}^{2} L_{i} and assume that h is a grading element of L so that ad h |_{L_{i}}= i Id_{L_{i}}. The grading of L gives rise to an associated grading Der(L) = ⨁_{i∈Z} Der(L)_{i} of the derivation algebra Der(L) of L such that Der(L)_{i} = {D ∈ Der(L) | D(L_{j}) ⊆ L_{i+j} for all j}. This follows from the fact that Der(L)_{i} is the eigenspace of ad h corresponding to the eigenvalue i. Also, it is clear that Der(L)_{i} = (0) whenever |i| > 4, and that ad L_{i} ⊆ Der(L)_{i}.

8.2 For such a 5-graded Lie algebra L we consider the JKP 𝒫(L) associated with it (see Theorem 4.5). As a vector space L = 𝒫(L) ⊕ L_{0} . It follows from the definition of 𝒫(L) that the restriction map ρ : D ↦ D|_{𝒫(L)} , for D ∈ Der(L)_{0} , determines a Lie homomorphism ρ : Der(L)_{0} → 𝔰𝔱𝔯(𝒫(L)). Moreover, ρ is an embedding if 𝒫(L) generates L.

8.3 We assume now that the 5-graded Lie algebra L = ⨁_{i=−2}^{2} L_{i} has a realization as in Theorem 5.24,

L = 𝔏(P, ℋ) = P_{−} ⊕ ({P, P }/ℋ) ⊕ P_{+},

where P = 𝒫(L) a JKP and ℋ is a subspace of HF(P). Since 𝒫(L) generates L, the map ρ is an embedding. To describe the image of Der(L)_{0} , recall from (P6) and (P7) that there is a representation of 𝔰𝔱𝔯(P) on {P, P }, where the action of T ∈ 𝔰𝔱𝔯(P) is given by

\begin{aligned}T \{a, b\} & = \{T a, b\} + \{a, T b\}, \\ T \{x, y\} & = \{T x, y\} + \{x, T y\} \end{aligned}

for {a, b}, {x, y} ∈ {P, P }. We define

(8.4)  𝔰𝔱𝔯(P, ℋ) = {T ∈ 𝔰𝔱𝔯(P) | T (ℋ) ⊆ ℋ}.

Lemma 8.5. ρ(Der(L)_{0}) = 𝔰𝔱𝔯(P, ℋ).

Proof. Consider an arbitrary element D ∈ Der(L)_{0} as an endomorphism of L_{0} = {P, P }/ℋ. Then for all a ∈ J_{+} and b ∈ J_{−},

\begin{aligned}D(\{a, b\} + ℋ) & = D([a, b]) = [Da, b] + [a, Db] = \{Da, b\} + \{a, Db\} + ℋ \\ & = ρ(D)\{a, b\} + ℋ, \end{aligned}

and similarly D({x, y}+ℋ) = ρ(D){x, y}+ℋ for all x ∈ M_{+} and y ∈ M_{−}. This implies that D is the map induced by ρ(D) on the quotient space {P, P }/ℋ and therefore ρ(D) ∈ 𝔰𝔱𝔯(P, ℋ).

The reverse inclusion follows from the fact that every element T ∈ 𝔰𝔱𝔯(P, ℋ) induces a derivation of L = P ⊕ ({P, P }/ℋ).

Theorem 8.6. Let L be a BC_{1} -graded Lie algebra. Then the Lie algebra Der(L) of derivations of L has a Z-grading Der(L) = ⨁_{i∈Z} Der(L)_{i} induced from the 5-grading on L, with Der(L)_{i} = (0) if |i| > 2, and with Der(L)_{i} = ad L_{i} for i = ±1, ±2. Moreover, (i) when L has type D_{1} , then L = 𝔏(P, ℋ) = J_{−} ⊕ M_{−} ⊕ ({P, P }/ℋ) ⊕ M_{+} ⊕ J_{+} for some JKP P = 𝒫(L) = (J, M) and some subspace ℋ of HF(P), and Der(L)_{0} ≅ 𝔰𝔱𝔯(P, ℋ);

(ii) when L has type C_{1} , then L = 𝒦(𝔍, 𝔐, ℋ) for some 𝔍-ternary algebra 𝔍𝔐 and some subspace ℋ of HF(𝔍, 𝔐), and Der(L)_{0} ≅ 𝔰𝔱𝔯(_{𝔍}𝔐, ℋ)

=^{def} {T ∈ 𝔰𝔱𝔯(_{𝔍}𝔐) | T (ℋ) ⊆ ℋ};

(iii) when L has type B_{1} , then L = 𝒦(X,^{−}, ℋ) for some structurable algebra (X,^{−}) and some subspace ℋ of HF(X,^{−}), and Der(L)_{0} ≅ 𝔰𝔱𝔯(X,^{−}, ℋ)

=^{def} {T ∈ 𝔰𝔱𝔯(X,^{−}) | T (ℋ) ⊆ ℋ}.

Proof. It is enough to argue that Der(L) ⊆ Der(L)_{0} + ∑_{i≠0} ad L_{i}. Suppose D ∈ Der(L), and let h be a grading element of L. Then D(h) = ∑_{i=−2}^{2} x_{i} for some x_{i} ∈ L_{i}. For every y_{j} ∈ L_{j},

jD(y_{j}) = D([h, y_{j}]) = [ ∑_{i=−2}^{2} x_{i}, y_{j}] + [h, D(y_{j})].

The (i + j)th-component of this equality is iD(y_{j}) = [x_{i}, y_{j}]. Therefore the ithcomponent of D acts as i^{−1} ad x_{i} ∈ ad L_{i} for i ≠ 0, and D − ∑_{i≠0} i^{−1} ad x_{i} ∈ Der(L)_{0} . This, together with Lemma 8.5, gives (i).

Now Proposition 6.17 and Lemma 5.4 imply that for a J-ternary algebra _{𝔍}𝔐, the space {_{𝔍}𝔐,_{𝔍}𝔐} is a module over the Lie algebra 𝔰𝔱𝔯(_{𝔍}𝔐) ≅ 𝔰𝔱𝔯(𝒫(𝔍, 𝔐)). The results in (i) applied to the JKP 𝒫(𝔍, 𝔐) now produce the desired conclusions.

Analogously, for a structurable algebra (X,^{−}) the space {X, X} is a module over the Lie algebra 𝔰𝔱𝔯(X,^{−}) ≅ 𝔰𝔱𝔯(𝒫(X,^{−})) by Propositions 7.12 and 7.14, so that the result follows from (i) with the JKP 𝒫(X,^{−}).

9. Invariant bilinear forms on BC_{1} -graded Lie algebras

9.1 Our aim in this section is to describe the invariant bilinear forms on BC_{1} - graded Lie algebras in terms of invariant forms on their coordinate JKP. Recall from (3.11-3.12) that an invariant form on a JKP P = (J, M) can be viewed as a symmetric bilinear form g on the vector space J_{−} ⊕ M_{−} ⊕ M_{+} ⊕ J_{+} and that the transformations δ(a, b) and υ(x, y) on that space are skew-symmetric relative to this form.

Invariant forms on J-ternary algebras

9.2 As a J-ternary algebra _{𝔍}𝔐 is just a special case of JKP, the notion of an invariant form on _{𝔍}𝔐 should come from the notion of an invariant form on a JKP, but we would hope to have a simpler description in this particular case. Suppose P = (𝔍, 𝔐) is the JKP associated to _{𝔍}𝔐 as described in Theorem 6.12, and let ( , ) : J_{+} × J_{−} → F, ( , ) : M_{+} × M_{−} → F be an invariant form on P. One obvious simplification comes from the fact that J_{+} = 𝔍 = J_{−} and M_{+} = 𝔐 = M_{−}, so we can set ≺ a, b ≻= (a_{+}, b_{−}) and ≺ x, y ≻= (x_{+}, y_{−}) to obtain bilinear forms on 𝔍 and 𝔐 respectively. Here we use the notation x_{σ} for an element x ∈ 𝔍 ∪ 𝔐 to mean that x is considered as an element of J_{σ} ∪ M_{σ}.

Since b_{−} = \frac{1}{2}D_{𝔢_{−},b_{+}}𝔢_{−}, identity (IF1) (in 3.11) implies that ≺ a, b ≻= (a_{+}, b_{−}) = (a_{+}, \frac{1}{2}D_{𝔢_{−},b_{+}}𝔢_{−}) = (\frac{1}{2}D_{b_{+},𝔢_{−}}a_{+}, 𝔢_{−}) = ((ab)_{+}, 𝔢_{−}) and therefore ≺ a, b ≻ is symmetric.

On the other hand it follows from (IF5) that ≺ x, y ≻= (x_{+}, y_{−}) = (x_{+}, 𝔢_{−} ∘y_{+}) = −(y_{+}, 𝔢_{−} ∘x_{+}) = − ≺ y, x ≻, that is ≺ x, y ≻ is skew-symmetric.

Identity (IF1) can be rewritten as

(9.3)  ≺ ab, c ≻=≺ a, bc ≻,

while (IF2) and (IF3) can be expressed as

(9.4)  ≺ ⟨x, y, z⟩, w ≻ = − ≺ z, ⟨y, x, w⟩ ≻  and
(9.5)  ≺ ⟨x, z⟩, b ≻ =≺ x, b ∘ z ≻ .

Then (IF4) follows from the fact that ≺ x, y ≻ is skew-symmetric. These considerations justify the following

Definition 9.6. An invariant form on a J-ternary algebra _{𝔍}𝔐 consists of a symmetric bilinear form ≺ , ≻: 𝔍 × 𝔍 → F and a skew-symmetric bilinear form ≺ , ≻: 𝔐 × 𝔐 → F for which identities (9.3)-(9.5) hold.

Remark. Note that (9.3) says that ≺ , ≻: 𝔍×𝔍 → F is an invariant symmetric bilinear form relative to the Jordan product on 𝔍.

Every invariant form ≺ , ≻ on a 𝔍-ternary algebra _{𝔍}𝔐 defines a unique invariant form ( , ) on the JKP P = 𝒫(𝔍, 𝔐) by (a_{+}, b_{−}) =≺ a, b ≻ and (x_{+}, y_{−}) =≺ x, y ≻.

Invariant forms on structurable algebras

Definition 9.7. A bilinear form f on a structurable algebra (X,^{−}) is symmetric provided f(x, y) = f(y, x) for all x, y ∈ X, and it is invariant if

(9.8)  f(x̄, ȳ) = f(x, y)  and  f(xz, w) = f(z, x̄w),

for all x, z, w ∈ X.

Lemma 9.9. If f is a symmetric invariant form on the structurable algebra (X,^{−}), then there is an invariant form f̂ defined on the JKP 𝒫(X,^{−}) by f̂|_{X×X} = f|_{X×X} and f̂|_{S×S} = f|_{S×S}.

Proof. The fact that f̂ is invariant can be verified directly, or it can be found in [31, Eqs. (8-9)] and [3, Eq. (42)].

This brings us to the main result on forms, Theorem 9.10.

(i) Let L be a BC_{1} -graded Lie algebra of type D_{1} : L = 𝔏(P, ℋ) where P = (J, M) is a JKP and ℋ ⊆ HF(P). If f is an invariant form on P, then there is a unique symmetric invariant form g on L such that

(9.11)  g(a, b) = f(a, b) and g(x, y) = f(x, y)

for all a ∈ J_{+} , b ∈ J_{−}, x ∈ M_{+} , and y ∈ M_{−}. Conversely, for every symmetric invariant form g on L, there is a unique invariant form f on P satisfying (9.11).

(ii) Let L be a BC_{1} -graded Lie algebra of type C_{1} : L = 𝒦(𝔍, 𝔐, ℋ) for some 𝔍-ternary algebra _{𝔍}𝔐 and some subspace ℋ ⊆ HF(𝔍, 𝔐). If ≺ , ≻ is an invariant form on _{𝔍}𝔐, then there is a unique symmetric invariant form g on L such that

(9.12)  g(a, b) =≺ a, b ≻ and g(x, y) =≺ x, y ≻

for all a, b ∈ 𝔍, x, y ∈ 𝔐. Conversely, for every symmetric invariant form g on L, there is a unique invariant form ≺ , ≻ on _{𝔍}𝔐 satisfying (9.12).

(iii) Let L be a BC_{1} -graded Lie algebra of type B_{1} : L = 𝒦(X,^{−}, ℋ) for a structurable algebra (X,^{−}) and a subspace ℋ ⊆ HF(X,^{−}). If f is a symmetric invariant form on (X,^{−}), then there is a unique symmetric invariant form g on L such that

(9.13)  g((x, 0), (y, 0) ) = f(x, y)

for all x, y ∈ X. Conversely, for every symmetric invariant form g on L, there is a unique symmetric invariant form f on (X,^{−}) satisfying (9.13).

Proof. (i) We can assume that P = 𝒫(L). Let f be an invariant form of P. Then there is an invariant symmetric bilinear form f′ on the Lie algebra {P, P } described in (5.12). It is easy to see that HF(P) is contained in the radical of this form. Hence there is an induced form f_{0} on L_{0} = {P, P }/ℋ. The identities in (5.12) guarantee that condition (4.11) holds. As a consequence, Lemma 4.10 implies that g(p + l_{0}, p′ + l′_{0}) = f(p, p′) + f_{0}(l_{0}, l′_{0}) is an invariant symmetric bilinear form on L. The uniqueness of g follows from the fact that L is generated by the subspaces L_{i}, i ≠ 0.

For the other direction, assume g is an invariant symmetric bilinear form on L and h is a grading element of L. Then for every l_{i} ∈ L_{i} and l_{j} ∈ L_{j}, one has ig(l_{i}, l_{j}) = g([h, l_{i}], l_{j}) = −g(l_{i}, [h, l_{j}]) = −jg(l_{i}, l_{j}), which implies that

g(L_{i}, L_{j}) = 0 unless i + j = 0.

Now by Lemma 4.10, the restriction f of g to P is an invariant form on the JKP P. Clearly it is unique and satisfies (9.11).

(ii) The proof in this case follows from (i) and 9.2.

(iii) If f is a symmetric invariant form on (X,^{−}), then by Lemma 9.9 there is an invariant form f̂ on 𝒫(X,^{−}). Part (i) implies that f̂ can be extended in a unique way on 𝔏(𝒫(X,^{−}), ϑ^{−1}(ℋ)) ≅ 𝒦(X,^{−}, ℋ) so that property (9.13) holds.

Now assume that g is a symmetric invariant form on L = 𝒦(X,^{−}, ℋ). Define a bilinear form f on X by f(x, y) = g((x, 0), (y, 0) ). In verifying f is symmetric and satisfies the identities in (9.8), we will use the 𝔰𝔩_{2} -triple ⟨e, f, h⟩ = ⟨(2, 0), 2{1, 1} + ℋ, (1, 0) ⟩ in L and the following equations,

(x̄, 0) = (Id − \frac{1}{2}F E)(x, 0), (x̄, 0) = (Id − \frac{1}{2}EF)(x, 0),

(x, 0) = (\frac{1}{3}F^{3}E − F^{2})(x, 0) = (\frac{1}{3}EF^{3} − F^{2})(x, 0),

where E = ad e, F = ad f, and H = ad h. All these relations except the last one are proved in [4] for the Lie algebra 𝒦(X,^{−}). In our situation they can be readily checked. The last identity follows from the equation [E, F^{3}] = 3F^{2}(H − 2Id), which is a corollary of the 𝔰𝔩_{2} -relations. From these equations and properties of g we have

f(x̄, ȳ) = g((x̄, 0), (ȳ, 0) ) = g(((Id − \frac{1}{2}F E)(x, 0), 0), (ȳ, 0) )

= g((x, 0), (Id − \frac{1}{2}EF)(ȳ, 0) ) = g((x, 0), (¯ȳ, 0) )

= f(x, y), and

f(x, y) = g((x, 0), (y, 0) ) = g((x, 0), (\frac{1}{3}F^{3}E − F^{2})(y, 0))

= g((\frac{1}{3}EF^{3} − F^{2})(x, 0), (y, 0)) = g((x, 0) , (y, 0))

= f(y, x).

Finally, the invariance of g implies f(V_{x,y}z, w) = f(z, V_{y,x}w). When y = 1, this amounts to

f(xz + zx − zx̄, w) = f(z, x̄w + wx̄ − wx).

If x̄ = x, it is immediate that f(xz, w) = f(z, xw) as desired.

If instead x̄ = −x, then f(xz + 2zx, w) = f(z, −xw − 2wx). Applying (9.8) to this equation and replacing z̄, w̄ with z, w, we see that f(zx+2xz, w) = f(z, −wx − 2xw). Combining the last two equations, we obtain the desired conclusion f(xz, w) = −f(z, xw).

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G. Benkart Department of Mathematics University of Wisconsin Madison, Wisconsin 53706 benkart@math.wisc.edu O. Smirnov Department of Mathematics College of Charleston Charleston, South Carolina 29424 smirnov@cofc.edu

Received October 2, 2001 and in final form December 18, 2001