Homology, Homotopy and Applications, vol.4(2), 2002, pp.219–225

GROWTH AND LIE BRACKETS IN THE HOMOTOPY LIE ALGEBRA

YVES FÉLIX, STEPHEN HALPERIN and JEAN-CLAUDE THOMAS

(communicated by Clas Löfwall)

Abstract Let L be an infinite dimensional graded Lie algebra that is either the homotopy Lie algebra π_{∗}(ΩX) ⊗ ℚ for a finite ndimensional CW complex X, or else the homotopy Lie algebra for a local noetherian commutative ring R (UL = Ext_{R}(Ik, Ik)) in which case put n = (embdim − depth)(R).

Theorem: (i) The integers λ_{k} = ∑_{q=k}^{k+n−2} dimL_{i} grow faster

than any polynomial in k.

(ii) For some finite sequence x_{1}, …, x_{d} of elements in L and some N, any y ∈ L_{⩾N} satisfies: some [x_{i}, y] ≠ 0.

To Jan–Erik Roos on his sixty–fifth birthday

1. Introduction

Let X be a simply connected CW complex of finite type. Then [16] its loop space homology, H_{∗}(ΩX; ℚ) is the universal enveloping algebra of the graded Lie algebra L_{X} = {(L_{X})_{i}}_{i⩾1} = π_{∗}(ΩX) ⊗ ℚ, equipped with the Samelson product. Similarly, if R is a commutative local noetherian ring with residue field Ik then [1], [17] Ext_{R}(Ik, Ik) is the universal enveloping algebra of a graded Lie algebra L_{R} = {L_{R}^{i}}_{i⩾1}. We call L_{X} (L_{R}) the homotopy Lie algebra of X (of R) and call e_{i}(X) = dim (L_{X})_{i} (or e_{i}(R) = dim L_{R}^{i}) the i^{th} deviation of X (or of R).

For finite complexes X and for all local rings R the hypothesis dim L < ∞ imposes very special conditions (in this case X is called ℚ-elliptic). For example, if X is ℚ-elliptic the H_{∗}(ΩX; ℚ) is a Poincaré duality algebra [12] while if L_{R} is finite dimensional then R is a complete intersection [10], [11]. Moreover, it is known (again for finite complexes X and for any R) that

• If dim L_{X} < ∞ and dim L_{R} < ∞ then e_{i}(X) = 0, i ⩾ 2 dim X, [9] and e_{i}(R) = 0, i ⩾ 3, [10], [11].

• If dim L_{X} = ∞ and dim L_{R} = ∞ then for some K > 0, C > 1,

Received January 8, 2001, revised September 17, 2001; published on July 12, 2002. 2000 Mathematics Subject Classification: 55P62, 55P35, 17B70, 16L99. Key words and phrases: Finite CW complex - Local ring - Homotopy Lie algebra - Depth. \begin{matrix}c \\ ◯ \end{matrix} 2002, Yves Félix, Stephen Halperin and Jean-Claude Thomas. Permission to copy for private use granted.

∑_{i=1}^{k} e_{i}(X) ⩾ KC^{k}, k ⩾ dim X − 1, [3] and ∑_{i=1}^{k} e_{i}(R) ⩾ KC^{k}, k ⩾ 1, [2].

• If dim L_{X} = ∞ and dim L_{R} = ∞ then

∑_{i=k}^{k+dim X−2} e_{i}(X) > 0, all k ⩾ 1, [14] and e_{k}(R) > 0, all k ⩾ 1, [13].

• If dim L = ∞ (L = L_{X} or L_{R}) then for all x ∈ L of sufficiently large even degree there is some y = y(x) ∈ L such that (adx)^{k}y ≠ 0, k ⩾ 1, [7].

These results motivate/provide evidence for the two following main conjectures, due to some combination of Avramov - Félix - Halperin - Thomas.

Conjecture 1. If X is finite dimensional, not ℚ-elliptic, and if R is not a complete intersection then for some K > 0, C > 1:

∑_{i=k}^{k+dim X−2} e_{i}(X) ⩾ KC^{k}, k ⩾ 1, and e_{k}(R) ⩾ KC^{k}, k ⩾ 1.

Conjecture 2. If X is finite dimensional, not ℚ-elliptic, and if R is not a complete intersection then L_{X} and L_{R} each contain a free Lie subalgebra on two generators.

This paper makes some progress towards these conjectures. For simplicity we adopt the following notation:

• X is a finite, non ℚ-elliptic, simply connected CW complex and R is a local noetherian commutative ring that is not a complete intersection.

• L is either L_{X} or L_{R}, and L_{even} is the sub Lie algebra of elements of even degree.

• n = n_{X} = dim X or n = n_{R} = (emb dim−depth)(R).

• e_{i} = e_{i}(X), or e_{i} = e_{i}(R).

• h = h_{X} = dim H^{∗}(X; ℚ), or h = h_{R} = dim H_{∗}(K^{R}), K^{R} denoting the Koszul

complex of R. Then, with the hypotheses and notation above, we establish

Theorem A.

(i) The integers λ_{k} = ∑_{i=k}^{k+n−2} e_{k} grow faster than any polynomial in k. In partic-

ular,

λ_{k} → ∞ as k → ∞.

(ii) Moreover, if L_{even} contains a maximal abelian sub Lie algebra of finite dimension then for some K > 0, C > 1,

λ_{k} ⩾ KC^{k}, k ⩾ 1.

Theorem B. There is a finite sequence x_{1}, …, x_{d} of elements in L and an integer N such that:

y ∈ L , deg y ⩾ N ⇒ some [x_{i}, y] ≠ 0 .

2. General remarks.

With X is associated the commutative graded differential algebra A_{P L}(X) whose Sullivan minimal model (ΛV, d) satisfies [18], [8]

H(ΛV, d) ≅ H^{∗}(X; ℚ) and e_{i}(X) = dim V^{i+1}, i ⩾ 1.

In particular, H^{i}(ΛV, d) = 0, i > n. Moreover, [18], [8] the differential d = ∑_{i⩾2} d_{i},

with d_{i} : V → Λ^{i}V. Finally, (ΛV, d_{2}) = C^{∗}(L_{X}) where for any graded Lie algebra E over Ik, C^{∗}(E) is the Cartan-Chevalley-Eilenberg-Quillen complex, whose cohomology is Ext_{UE}(Ik, Ik).

Similarly, with R is associated its Koszul complex K^{R} which is connected by quasi-isomorphisms to a commutative graded chain algebra [2]. This in turn has a ’Sullivan model’ (ΛV, d) in which V = {V_{i}}_{i⩾1} and d decreases degrees by 1. Here we have H_{i}(ΛV, d) = H_{i}(K^{R}) = 0, i > n, and

e_{i}(R) = dim V_{i−1}, i ⩾ 2 .

Moreover (ΛV, d_{2}) = C^{∗}(L_{R}^{⩾2}).

Recall now that the depth of an augmented graded algebra A is the least m (or ∞) such that Ext_{A}^{m}(Ik, A) ≠ 0. We define the depth of a graded Lie algebra, E, to be the depth of its universal enveloping algebra (depth E = depth UE) and recall from [4] that

depth L_{X} ⩽ LScat(X) ⩽ n_{X} and depth L_{R} ⩽ n_{R} .  (2.1)

We shall make frequent use of the remark [4] that if I is an ideal in a graded Lie algebra E then

depth I ⩽ depth E .  (2.2)

Finally, since in both cases we have dim H(ΛV, d) = h < ∞, we can apply a result of Lambrechts:

Lemma 2.3 [15]. For all k sufficiently large, there is some l ∈ [k + 1, k + n − 1] such that dim V^{l} ⩾ dim V^{k} /hn.

In fact Lambrechts shows that dim V^{k} ⩽ h ∑_{i=1}^{n−1} dim V^{k+i}+ dim G_{k}, where G_{∗} ⊂ L

is the abelian ideal of Gottlieb elements. As noted in [4] this implies that G_{∗} is finite dimensional, and so the inequality of Lemma 2.3 holds for large k.

3. Proof of Theorem A.

(i) We prove this in the case that L_{even} contains an infinite dimensional abelian sub Lie algebra, E, since otherwise (i) will follow from (ii). For convenience, we abuse notation and write the degrees as subscripts.

Note that the sub Lie algebra F = E_{⩽k} ⊕ L_{>k} has finite codimension in L. Thus we can write F = I^{m} ⊂ I^{m−1} ⊂ ⋯ ⊂ I^{0} = L where each I^{k} is constructed from I^{k+1} by adding a single element of maximal degree. It follows that each I^{k} is a sub Lie algebra containing I^{k+1} as an ideal. In particular by (2.1) and (2.2).

depth F ⩽ depth L ⩽ n .

On the other hand F/L_{>k} is the abelian Lie algebra E_{⩽k} and UE_{⩽k} = Ik[E_{⩽k}] is a polynomial algebra. In particular, depth E_{⩽k} = dim E_{⩽k}, and there are constants 0 < c < C such that for any finitely generated UE_{⩽k}-module M, and some integer r(M),

ck^{r(M)} ⩽ ∑_{i⩽k} dim M_{i} ⩽ Ck^{r(M)}, k sufficiently large.

The integer r(M) is called the polynomial growth of M.

Now ([6];Theorem 4.1) asserts that for some q ⩽ n and some α ∈ T or_{q}^{UL_{>k}}(Ik, Ik) the module UE_{⩽k} · α has polynomial growth at least equal to (dim E_{⩽k}) − n. But the action of UE_{⩽k} in T or^{UL_{>k}}(Ik, Ik) is induced from the adjoint representation of E_{⩽k} in the complex (Λ^{C}(sL_{>k}), ∂) dual to C^{∗}(L_{>k}); here Λ^{C} denotes the free co-commutative coalgebra. In particular for some z ∈ (Λ^{C})^{q}sL_{>k}, UE_{⩽k} · z has polynomial growth at least equal to (dim E_{⩽k}) − n.

Since q ⩽ n this implies in turn that for some y ∈ L_{>k},

poly growth (UE_{⩽k} · y) ⩾ \frac{dim E_{⩽k}}{n} − 1 .

Fix some r > 0 and choose k so that dim E_{⩽k} ⩾ (n + 1)r. Then poly growth (UE_{⩽k} · y) ⩾ r. It follows that there are r elements x_{1}, …, x_{r} ∈ E_{⩽k} such that Ik[x_{1}, …, x_{r}] →^{≅} Ik[x_{1}, …, x_{r}] · y. Choosing d_{i} so that the x_{i}^{d_{i}} all have the same degree d we see that

dim L_{kd+deg y} ⩾ λk^{r} ⩾ μ((k + 1)d + deg y)^{r} , k ⩾ 2 ,  (3.1)

for some positive constants λ and μ. Now, for k sufficiently large, repeated applications of Lemma 2.3 give an infinite sequence of integers i_{1} < i_{2} < ... such that i_{1} = kd + degy, and

i_{s+1} ⩽ i_{s} + n − 1 and dim L_{i_{s}} ⩾ \frac{μ((k + 1)d + deg y)^{r}}{(nh)^{s}} , s ⩾ 1 .

It follows at once that (provided k is sufficiently large)

∑_{j=q}^{q+n−2} dim L_{j} ⩾ \frac{μ}{(nh)^{d}} q^{r} , deg y + kd ⩽ q ⩽ deg y + (k + 1)d .

Since both sides of the equation are independent of k this establishes (i) in the presence of an infinite dimensional abelian subalgebra. (ii) Let E = ⊕_{i=1}^{r}Ikx_{i} be a maximal abelian sub Lie algebra of L_{even}. Give L_{even} the decreasing filtration defined by F^{0} = L_{even}, and F^{i} = {y ∈ L_{even} | [x_{j}, y] = 0 , 1 ⩽ j ⩽ i}. The maximality of E implies that F^{r} = 0. Choose graded subspaces V^{i} ⊂ L_{even} such that F^{i−1} = V^{i} ⊕ F^{i}, and choose integers d_{1}, ..., d_{r} so that d_{1}deg x_{1} = ... = d_{r}deg x_{r} = d. Then for all q and all k

(adx_{1})^{qd_{1}} ⊕ ... ⊕ (adx_{r})^{qd_{r}} : V_{2k}^{1} ⊕ ... ⊕ V_{2k}^{r} ⟶ L_{2k+qd}

is injective ; i.e. dim L_{2k} ⩽ dim L_{2k+qd}, k ⩾ 0, q ⩾ 0.

On the other hand, a simple extension of the argument in ([8]; Chapter 33) gives an infinite sequence of even integers i_{1} < i_{2} < ... such that i_{s+1} ⩽ n^{2}i_{s}, s ⩾ 1, and constants a > 0, D > 1 such that dim L_{i_{s}} ⩾ aD^{i_{s}}, s ⩾ 1. Now application of Lambrecht’s lemma 2.3 gives (ii) in the same way it completed the proof of (i).

□

4. Proof of Theorem B.

As recalled in §2, L has finite depth. This means that Ext_{UL}(Ik, UL) ≠ 0, and in [5] it is shown that for some finitely generated sub Hopf algebra G the restriction Ext_{UL}(Ik, UL) → Ext_{G}(Ik, UL) is non zero. Suppose G is generated in degrees less than or equal to n, and denote E = L_{⩽n}. Then the restriction Ext_{UL}(Ik, UL) → Ext_{G}(Ik, UL) factors through Ext_{UE}(Ik, UL), and so the restriction Ext_{UL}(Ik, UL) → Ext_{UE}(Ik, UL) is non zero. In particular, E has finite depth. The adjoint action of E in L defines a representation of UE in L, and Theorem B is a corollary of

Theorem C. For some N and all y ∈ L_{⩾N} the graded vector space UE · y grows faster than any polynomial.

Proof. Let Z ⊂ L be the subspace of elements z such that UE · z grows at most polynomially (i.e. for some constant c > 0 and some r, dim [UE · z]_{k} ⩽ ck^{r}, k ⩾ 1. Since UE · [z, w] ⊂ [UE · z, UE · w] it follows that Z is a sub Lie algebra of L, stable under the adjoint representation of E.

In particular, if x ∈ Z ∩ E then UE.x is an ideal in E of at most polynomial growth. Since depth UE.x ⩽ depth E < ∞ (by 2.2) if follows from ([6]; Theorem B) that UE.x is finite dimensional . Thus Z ∩ E is an ideal in E that is the union of finite dimensional ideals. Since L = L_{⩾1} these finite dimensional ideals are solvable and their sum Z ∩ E is then itself finite dimensional [4].

Thus Z_{⩾q} ∩E = 0 (some q) and E ⊕Z_{⩾q} is itself a sub Lie algebra of L. Moreover the composite

Ext_{UL}(Ik, UL) → Ext_{U(E⊕Z_{⩾q})}(Ik, UL) → Ext_{UE}(Ik, UL)

is non-zero. But Ext_{U(E⊕Z_{⩾q})}(Ik, UL) is the cohomology of the complex (∧(sE)^{∗} ⊗ ∧(sZ_{⩾q})^{∗} ⊗ UL, d), and a simple ‘filtration argument’ shows that the restriction to (∧(sE)^{∗} ⊗ UL, d) is zero in cohomology unless for some a ∈ UL , 1 ⊗ a is a cocycle in the quotient complex (∧(sZ_{⩾q})^{∗} ⊗ UL, d). This can only occur when Z_{⩾q} is finite dimensional and concentrated in odd degrees [4].

Thus Z itself is finite dimensional and it suffices to choose N so that Z is concentrated in degrees < N.

Homology, Homotopy and Applications, vol. 4(2), 2002 224

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This article may be accessed via WWW at http://www.rmi.acnet.ge/hha/ or by anonymous ftp at

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Yves Félix Institut de Mathématiques, Université de Louvain-La-Neuve, B-1348 Louvain-La-Neuve, Belgium.

Stephen Halperin

College of Computer, Mathematical and Physical Sciences, University of Maryland, College Park, MD 20742-3281, USA.

Jean-Claude Thomas thomas@tonton.univ-angers.fr

Facultd́es Sciences Université d’Angers, 49045 bd Lavoisier, Angers, France.