Journal of Lie Theory Volume 10 (2000) 455–461 \begin{matrix}c \\ ◯ \end{matrix} 2000 Heldermann Verlag Berlin

Geodesic loops

Ágota Figula^{∗}

Communicated by K. Strambach

Abstract. Using new tangential objects, which we call Λ-algebras, we obtain for geodesic loops (L, ∇, e) statements that are analogous to Lie’s theorems. We prove that any geodesic loop L defines in the tangential space T_{e}L a unique Λ-algebra, and that to any finite dimensional real Λ-algebra F there exists a geodesic loop L, the Λ-algebra of which is isomorphic to F. Geodesic loops which have isomorphic Λ-algebras, are locally isomorphic.

If the local loop L is diassociative then to the Λ-algebra of L there corresponds a subseries of the Hausdorff-Campbell formula with respect to the binary Lie algebra belonging to L.

Introduction

A.I. Malcev perceived that the fundamental ideas of Sophus Lie to assign to any local Lie group G a tangential object -its Lie algebra- which determines G in a unique way can be extended to non-associative structures. In particular with any differentiable Moufang loop L is associated a Malcev algebra which determines L uniquely up to coverings (cf. [9] Theorem 4.4). Moreover, in the mean time the theory of Lie Moufang loops reached the same level as the theory of Lie groups (cf. [3], [5], [8]). The most general class of local analytical loops for which we know that they are determined by their tangential objects -by their Bol algebras- in a unique way are the Bol loops (cf. [7] pp. 424-425 ). In general, the differentiable loops have as tangential objects the Akivis algebras, but an Akivis algebra corresponds to a variety of local differentiable loops the algebraic properties of which show greate differences (cf. [2] pp. 240-241).

The aim of this paper is to show that there exist suitable tangential objects for very wide classes of local analytical loops L which determines L in a unique way; concretely we demonstrate this for the class of geodesic loops which respect to linear connections the curvature of which is zero; these tangential objects have in general infinitely many binary operations. Since a differentiable local loop is strongly left alternative if and only if it can be represented as the opposite loop of a geodesic loop with respect to a linear connection with curvature zero, the local

ISSN 0949–5932 / $2.50 \begin{matrix}c \\ ◯ \end{matrix} Heldermann Verlag analytic Bol loops are a proper subclass of geodesic loops considered in our paper (cf. [11] Proposition 4.7).

∗Partially supported by a Grant 323 PPP-Ungarn of DAAD

The tangential object T_{e}L of a geodesic loop L with respect to a linear connection with vanishing curvature is called a Λ-algebra. The infinitely many binary operations Λ_{(p,1)}(x, y) : T_{e}L × T_{e}L → T_{e}L are linear in y and p-homogeneous in x; moreover, they satisfy certain symmetry conditions. In the case that the local loop L is diassociative to the Λ-algebra of L there corresponds a subseries of the Hausdorff-Campbell formula with respect to the binary Lie algebra belonging to L.

I would like to express my gratitude to my teacher P.T. Nagy for his guidance and encouragement.

1. Geodesic loops and their Λ-algebras

Definition 1.1. Let L be an analytic manifold and B an open neighbourhood of the identity e of L. Let ∗, /, \ : B×B → L be analytic mappings. The manifold L is called a local analytic loop if the analytic mappings ∗, /, \ : B×B → L satisfy the following identities:

1. x ∗ e = e ∗ x = x/e = e\x = x for all x ∈ B.
2. (x/y) ∗ y = x, y ∗ (y\x) = x,
(x ∗ y)/y = x, y\(y ∗ x) = x for all x, y ∈ B.

provided the left sides of these equations are defined.

Definition 1.2. Let L be a differentiable manifold equipped with an affine connection ∇ and let e be a point in L. A local analytic loop (L, ∇, e) is called a geodesic loop if in a normal neighbourhood U of e ∈ L the multiplication of L is given by

x ∗ y = exp_{y} ∘ τ_{e,y} ∘ exp_{e}^{−1}(x)

for all x, y ∈ U, where τ_{e,y} denotes the parallel translation T_{e}L → T_{y}L along the unique geodesic segment, contained in U, between e and y (cf. [4], p. 160 and [7], p. 369).

Definition 1.3. Let V be an n-dimensional real or complex vector space. V is called a Λ-algebra if there exists for every natural number p an algebraic map Λ_{(p,1)} : V × V → V such that the following properties are satisfied:

1. Λ_{(p,1)}(x, λy) = λΛ_{(p,1)}(x, y) for all λ ∈ K and x, y ∈ V.
2. Λ_{(p,1)}(λx, y) = λ^{p}Λ_{(p,1)}(x, y) for all λ ∈ K and x, y ∈ V.
3.  ∑_{p=1}^{∞} Λ_{(p,1)}(x, y)  converges on a neigbourhood N of 0 in V.

Remark 1.4. The coordinate functions Λ_{(p,1)}^{i}(x, y), i = 1, …, n, of Λ_{(p,1)}(x, y) are homogeneous polynomials of degree p + 1 which are linear in the coordinates of y and homogeneous polynomials of degree p in the coordinates of x.

The Λ-algebras are suitable tangential objects for geodesic loops (L, ∇, e) for which the curvature tensor is 0, in order to obtain analogous statements to Lie’s theorems.

Theorem 1.5. (Lie’s First Theorem) Any geodesic loop L defines in the tangential space T_{e}L a unique Λ-algebra.

Proof. The 1-parameter subgroups of a geodesic loop (L, ∇, e) are geodesic lines through the identity e of L with respect to the linear connection ∇. Hence the canonical coordinate system for L (which is determined by the 1-parameter subgroups) is a normal coordinate system of the connection ∇ in a normal neighbourhood U ⊂ ℝ^{n}. Since the curvature of ∇ is zero every parallel translation with respect to ∇ can be given in U by the independent parallel vector fields A_{1}(x), …, A_{n}(x), (i = 1, …, n), where A_{i}(0) = e_{i} yield the canonical basis of ℝ^{n}. The parallel translation τ_{e,y} is determined by e_{i} ↦ A_{i}(y) (i = 1, …, n). Hence

the map ∑_{i=1}^{n} ξ^{i}e_{i} ↦ ∑_{i=1}^{n} A_{i}(y)ξ^{i} gives the tangent map T_{e}ρ_{y} for the right trans-

lation ρ_{y} : x → x ∗ y. The differential equations of the coordinate functions y^{i}(t)

for the geodesic line y(t) with the initial values y(0) = y_{0} and y′(0) = ∑_{j=1}^{n} A_{j}(y_{0})ξ^{j}

have the form

y′^{i}(t) = ∑_{j=1}^{n} A_{j}^{i}(y(t))ξ^{j},  (1)

where A_{j}^{i} denote the coordinate functions of the vector fields A_{j} and ξ^{j} are the coordinates with respect to the basis A_{j}(y_{0}). Since in the coordinate system of U the geodesic lines are the euclidean lines (ξ^{1}, …, ξ^{n})t, t ∈ ℝ, equation (1) implies that the coordinate functions A_{j}^{i}(x) of A_{j} satisfy the identities

ξ^{i} = ∑_{j=1}^{n} A_{j}^{i}((ξ^{1}, …, ξ^{n})t)ξ^{j}.

The power series of the coordinate functions A_{j}^{i}(x) through the point (0, …, 0) can be written as

ξ^{i} = ∑_{j=1}^{n}[A_{j}^{i}(0) + ∑_{p=1}^{∞} \frac{1}{p!} ∑_{j_{1},... ,j_{p}=1}^{n} \frac{∂^{p}A_{j}^{i}}{∂x^{j_{1}} … ∂x^{j_{p}}} (0)ξ^{j_{1}} … ξ^{j_{p}}]ξ^{j}.

Since A_{j}^{i}(0) = δ_{j}^{i} we obtain

ξ^{i} = ξ^{i} + ∑_{p=1}^{∞} \frac{1}{p!} ∑_{j_{1},... ,j_{p}=1}^{n} \frac{∂^{p}A_{j}^{i}}{∂x^{j_{1}} … ∂x^{j_{p}}} (0)ξ^{j_{1}} … ξ^{j_{p}}ξ^{j_{p+1}}.

Consequently the coefficients of this expansion satisfy

∑_{σ∈ℤ_{p+1}} \frac{∂^{p}A_{j(p+1)σ}^{i}}{∂x^{j_{1σ}} … ∂x^{j_{pσ}}} (0) = 0,  (2)

where ℤ_{p+1} is the cyclic permutation group of order p + 1.

In the tangential space of the identity e ∈ L we define now the binary operations Λ_{(p,1)} as follows:

Λ_{(1,1)}(x, y) = ( ∑_{j=1,h_{1}=1}^{n} \frac{∂A_{j}^{i}}{∂x^{h_{1}}} (0)x^{h_{1}}y^{j})_{i=1}^{n}

Λ_{(2,1)}(x, y) = ( ∑_{j=1,h_{1}=1,h_{2}=1}^{n} \frac{1}{2!} \frac{∂^{2}A_{j}^{i}}{∂x^{h_{1}}∂x^{h_{2}}} (0)x^{h_{1}}x^{h_{2}}y^{j})_{i=1}^{n}

Λ_{(p,1)}(x, y) = ( ∑_{j=1,h_{1}=1,... ,h_{p}=1}^{n} \frac{1}{p!} \frac{∂^{p}A_{j}^{i}}{∂x^{h_{1}} … ∂x^{h_{p}}} (0)x^{h_{1}} … x^{h_{p}}y^{j})_{i=1}^{n}.

The i-th coordinate of the vector Λ_{(p,1)}(x, y) is

Λ_{(p,1)}^{i}(x, y) = ∑_{j=1,h_{1}=1,... ,h_{p}=1}^{n} \frac{1}{p!} \frac{∂^{p}A_{j}^{i}}{∂x^{h_{1}} … ∂x^{h_{p}}} (0)x^{h_{1}} … x^{h_{p}}y^{j}.

This function is linear in the coordinates of y since it is depending only of the sum of the coordinates of y. Hence Λ_{(p,1)}(x, y) is linear in y. We have

Λ_{(p,1)}^{i}(λx, y) = ∑_{j=1,h_{1}=1,... ,h_{p}=1}^{n} \frac{1}{p!} \frac{∂^{p}A_{j}^{i}}{∂x^{h_{1}} … ∂x^{h_{p}}} (0)λx^{h_{1}} … λx^{h_{p}}y^{j} =

λ^{p}Λ_{(p,1)}^{i}(x, y).

Hence the second condition for a Λ-algebra is satisfied. The coordinate functions of the product (x^{1}, …, x^{n})∗(y^{1}, …, y^{n}) are given by A_{j}^{i}(y) and its partial derivaties

(cf.[10] pp.309-312). Since L is analytic loop and the series ∑_{p=1}^{∞} Λ_{(p,1)}(x, y) is a

subseries of the power expansion of the product x ∗ y the condition 3. in the definition of a Λ-algebra is valid.

Theorem 1.6. (Lie’s Third Theorem) To any finite dimensional real Λ-algebra F there exists a geodesic loop L, the Λ-algebra of which is isomorphic to F. The geodesic loops which have isomorphic Λ-algebras, are locally isomorphic.

Proof. Let F be an n-dimensional real Λ-algebra. The convergent series

∑_{p=1}^{∞} Λ_{(p,1)}(x, y) determines a function f = (f^{1}, …, f^{n}) from F × F into F. The

function f is linear in y since the operations Λ_{(p,1)}(x, y) are linear in y. Hence the i-th coordinate function f^{i}(x, y) has the form

f^{i}(x^{1}, …, x^{n}, y^{1}, …, y^{n}) = ∑_{j=1}^{n} f_{j}^{i}(x)y^{j}

with suitable real analytic functions f_{j}^{i}. If we develop the functions f_{j}^{i} into Taylor series in the point zero we obtain

f_{j}^{i}(x) = ∑_{p=1}^{∞} \frac{1}{p!} ∑_{h_{1}=1,... ,h_{p}=1}^{n} \frac{∂^{p}f_{j}^{i}}{∂x^{h_{1}} … ∂x^{h_{p}}} (0)x^{h_{1}} … x^{h_{p}}.

Since the real analytic function f is described uniquely by a convergent series we can write

Λ_{(p,1)}(x, y) = ( ∑_{j=1,h_{1}=1,... ,h_{p}=1}^{n} \frac{1}{p!} \frac{∂^{p}A_{j}^{i}}{∂x^{h_{1}} … ∂x^{h_{p}}} (0)x^{h_{1}} … x^{h_{p}}y^{j})_{i=1}^{n},

where A_{j}^{i} = f_{j}^{i}. The coordinate functions A_{j}^{i} determine the vector fields A_{j} which are defined as A_{j} : y → (A_{j}^{1}(y), …, A_{j}^{n}(y)). Considering F as the affine space

ℝ^{n} we can assign to any point y of F a tangent space T_{y}F = ∑_{i=1}^{n} η^{i}A_{i}(y), η_{i} ∈ ℝ

and form the tangent bundle T F. Any vector v ∈ T_{0}F, v = ∑_{i=1}^{n} η^{i}e_{i}, determines

a vector field Y_{v} putting Y_{v}(y) = ∑_{i=1}^{n} η^{i}A_{i}(y).

We define now a covariant derivation ∇, such that the vector fields A_{i} become parallel vector fields with respect to ∇. Hence in the expression

∇_{X}Y = ∇_{X}(∑_{i=1}^{n} Y^{i}A_{i}) = ∑_{i=1}^{n}(X(Y)^{i})A_{i} + ∑_{i=1}^{n} Y^{i}∇_{X}A_{i}

we put ∑_{i=1}^{n} Y^{i}∇_{X}A_{i} = 0 for any vector field X. For the curvature tensor R(X, Z)Y with respect to ∇ we have

R(X, Z)Y = ∇_{X}∇_{Z}(∑_{i=1}^{n} Y^{i}A_{i}) − ∇_{Z}∇_{X}(∑_{i=1}^{n} Y^{i}A_{i}) − ∇_{[X,Z]}(∑_{i=1}^{n} Y^{i}A_{i}) =

∇_{X}(∑_{i=1}^{n}(Z(Y^{i}))A_{i}) − ∇_{Z}(∑_{i=1}^{n}(X(Y^{i}))A_{i}) − ∑_{i=1}^{n}([X, Z](Y^{i}))A_{i} =

X(∑_{i=1}^{n}(Z(Y^{i}))A_{i}) − Z(∑_{i=1}^{n}(X(Y^{i}))A_{i}) − ∑_{i=1}^{n}([X, Z](Y^{i}))A_{i}.

Since [X, Z] = XZ − ZX we can write [X, Z](Y^{i}) = X(Z(Y^{i}) − Z(X(Y^{i})). It follows

R(X, Z)Y = X(∑_{i=1}^{n}(Z(Y^{i}))A_{i}) − Z(∑_{i=1}^{n}(X(Y^{i}))A_{i}) − ∑_{i=1}^{n}([X, Z](Y^{i}))A_{i} = 0.

The parallel translations τ_{0,y} are determined by τ_{0,y}(e_{i}) = A_{i}(y). Since the curvature tensor R with respect to ∇ is zero the parallel translations do not depend on the path. Let exp be the exponential map belonging to the linear connection ∇. We have exp_{y} : T_{y}F → F : exp_{y}(v_{y}) = γ(1) where γ is the geodesic curve on F with respect to ∇ satisfying the conditions γ(0) = y and γ′(0) = v_{y}. The multiplication x ∗ y = exp_{y} ∘ τ_{e,y} ∘ exp_{e}^{−1}(x) for x, y ∈ F defines on F a geodesic loop L with 0 as the identity. The Λ-algebra of L is uniquely determined by the parallel translations τ_{0,y} and hence by the vector fields A_{i}, i = 1, …, n, as well as by the derivaties of their coordinate functions.

A real algebra (C, [., .]) is called a binary Lie algebra if any two elements of C generate a Lie subalgebra.

To the Λ-algebras of Lie algebras and of binary Lie algebras there correspond subseries of the Campbell-Hausdorff-series. These subseries are determined by the Lie algebras or binary Lie algebras.

Theorem 1.7. Let C a binary Lie algebra and (b_{2n})_{n∈ℕ} the sequence of the even Bernoulli numbers. The operations Λ_{(p,1)} : C^{2} → C given by

\begin{aligned}Λ_{(1,1)}(X, Y) & = & \frac{1}{2}[X, Y] \\ Λ_{(2,1)}(X, Y) & = & \frac{1}{12}[X, [X, Y]] \\ Λ_{(2n,1)}(X, Y) & = & \frac{1}{(2n)!}b_{2n}[X, …, [X, Y], …] \\ Λ_{(2n+1,1)}(X, Y) & = & 0 \end{aligned}

define on the vector space C a Λ-algebra Λ(C).

Proof. The operations Λ_{(p,1)} are linear in Y . Moreover, we have

Λ_{(p,1)}(λX, Y) = λ^{p}Λ_{(p,1)}(X, Y)

for any λ ∈ ℝ. The series ∑_{p=1}^{∞} Λ_{(p,1)}(X, Y) converges since it is a subseries of the

Hausdorff-Campbell series (cf. [1] pp. 201-202).

An analytic loop L is called diassociative if every two elements generate a subgroup of L.

Theorem 1.8. The Λ-algebra F of a local diassociative analytic loop L = exp(F) coincides with the Λ-algebra Λ(C), where C is the binary Lie algebra belonging to L.

Proof. According to Malcev, theorems 1 and 2 (cf. [6] pp. 570-571), the tangential algebra of L is a binary Lie algebra C. The power series of the multiplication of L is the classical Hausdorff-Campbell series. Let Λ_{(p,1)}, p ∈ ℕ, be the operations of the Λ-algebra of L and Λ_{(p,1)}^{C}, p ∈ ℕ, be the operations for

the Λ-algebra Λ(C). Since ∑_{p=1}^{∞} Λ_{(p,1)}(x, y) is a subseries of the loop multiplication

and

∑_{p=1}^{∞} Λ_{(p,1)}(x, y) = ∑_{p=1}^{∞} Λ_{(p,1)}^{C}(x, y)

we have Λ_{(p,1)} = Λ_{(p,1)}^{C}.

Theorem 1.9. The local analytic loop L the Λ-algebra of which is Λ(C) for a binary Lie algebra C is diassociative and C is the tangential algebra of L in sense of Malcev.

Proof. Since Λ(C) is the Λ-algebra of L, the series

∑_{p=1}^{∞} Λ_{(p,1)}^{C}(x, y)  (3)

is a subseries of the series of the multiplication (x, y) ↦ x∗y : L×L → L such that

x ∗ y = x + y + ∑_{p,q=1}^{∞} Λ_{(p,q)}^{C}(x, y) and the operations Λ_{(p,q)}^{C} are uniquely determined

by Λ_{(p,1)}^{C} through a recursive formula (G) in (cf. [10] p. 311). Since (3) is a subseries of the classical Hausdorff-Campbell series too, hence the multiplication (x, y) ↦ x ∗ y is described by the classical Hausdorff-Campbell formula and there exists a diassociative loop having the same subseries (3) as L. The multiplication of this loop coincides with the multiplication of L and C is the binary Lie algebra in sense of Malcev (cf. [6] pp. 570-571).

Remark 1.10. Theorems 4 and 5 remain valid also for monoalternative analytic loops since any such loop is diassociative (cf. [12] ).

References

[1] Bourbaki, N., “Lie groups and Lie algebras, Chapter 1-3, ” Springer Verlag, 1989.

[2] Hofmann, K. H., and K. Strambach, Topological and analytical loops, in “Quasigroups and Loops: Theory and Applications” (Chein, O., Pflugfelder, H.O., and Smith, J.D.H.), Sigma Series in Pure Math. 8, Heldermann Verlag, 1990, 205–262.

[3] Kerdman, F. S., Moufang loops in the large (Russian), Algebra i Logika 18 (1979), 523–555.

[4] Kikkawa, M., Geometry of homogeneous Lie loops, Hiroshima Matematical Journal 5 (1975), 141–179.

[5] Kuzmin, E. N., The connection between Malcev algebras and analytic Moufang loops (Russian), Algebra i Logika 10 (1971), 3–22.

[6] Malcev, A. I., Analytical loops (Russian), Matematicheskii Sbornik 36 (1955), 569–576.

[7] Miheev, P. O., and L. V. Sabinin, Quasigroups and Differential Geometry, Chapter XII in “Quasigroups and Loops: Theory and Applications” (Chein, O., Pflugfelder, H.O., and Smith, J.D.H), Sigma Series in Pure Math. 8, Heldermann Verlag, 1990, pp. 357–430.

[8] Nagy, P. T., Extension of local loop isomorphisms, Monatshefte Math. 112 (1991), 221–225.

[9] —, Moufang loops and Malcev algebras, Journal of Lie Theory (=Seminar Sophus Lie) 3 (1992), 65–68.

[10] Nagy, P. T., and K. Strambach, Loops, their cores and symmetric spaces, Israel Journal of Mathematics 105 (1998), 285–322.

[11] — , “Sections in groups: A Lie Theory of smooth loops,” Preprint pp. 267.

[12] Sabinin, L. V., On the diassociativity of smooth monoalternative loops, Russian Mathematical Surveys 51 (1996), 747–749.

Mathematisches Institut der Universität Erlangen-Nürnberg Bismarckstr. 1 \frac{1}{2} , D-91054 Erlangen, Germany figula@mi.uni-erlangen.de Received October 21, 1999 and in final form January 12, 2000