Journal of Lie Theory Volume 12 (2002) 309–323 \begin{matrix}c \\ ◯ \end{matrix} 2002 Heldermann Verlag

On the Homology of Free Nilpotent Lie Algebras

Paulo Tirao

Communicated by D. Poguntke

Abstract. Explicit computations of the homology of some complex free nilpotent Lie algebras of small rank r, as modules over the general linear group GL(r, ℂ), are presented. A GL(r, ℂ)-Poincaré duality theorem and a stabilization theorem for r → ∞ are also proved.

The complex free N -step nilpotent Lie algebra of rank r, L(N, r), has a natural polynomial structure as a GL(r, ℂ)-module; this structure is induced to its homology groups. The description of these representations is an interesting problem.

The problem for the case of free 2-step nilpotent Lie algebras has been solved in [8], even though the same description can be deduced from the classical paper of Kostant [5] as shown in [4]. The GL(r, ℂ)-structure of the second homology group of any free nilpotent Lie algebra is also known, since it is isomorphic to H(N + 1), the subspace of (N +1)-brackets of the free Lie algebra and the GL(r, ℂ)-structure of the free Lie algebra was determined in [9].

We present in this paper some basic results and a list of explicit computations as a contribution to get a better perspective of the general problem and hoping they could inspire others to work on this problem. The computations have been done using Maple V.

In §2 we first prove a GL(r, ℂ)-version of the Poincaré duality for L(N, r), for all N. Recall that Poincaré duality holds for any finite dimensional nilpotent Lie algebra.

Next we investigate the relation between the homologies of two free N -step nilpotent Lie algebras of different rank. If F_{i}(r) is the family of Young diagrams describing the i-th homology group of L(N, r) as a GL(r, ℂ)-representation, then there are inclusions F_{i}(r) ↪ F_{i}(r + 1) ↪ F_{i}(r + 2) ↪ … and moreover there is an r_{0} such that F_{i}(r) ↪ F_{i}(r + 1) is a bijection for all r ≥ r_{0}.

Further restrictions on the Young diagrams that can occur in F_{i}(r) are deduced from a Gruenberg formula for Lie algebra homology.

In §3 we display all the homology groups computed as lists of Young diagrams. We computed the whole homology of the algebras L(III, 2,), L(IV, 2,), L(V, 2) and L(III, 3) and the groups H_{i}(L(III, r)), i = 1, …, 4, for all r.

At the moment we cannot explain our results, but we are confident that by

ISSN 0949–5932 / $2.50 \begin{matrix}c \\ ◯ \end{matrix} Heldermann Verlag explaining these particular cases one will get a deeper understanding of the general problem.

1. Preliminaries

Fix a natural number r ≥ 2, let X = {X_{1}, …, X_{r}} and let V_{r} be the ℂ-vector space spanned by X. Consider the tensor algebra

T (V_{r}) = ℂ ⊕ V_{r} ⊕ V_{r}^{⊗2} ⊕ ⋯ ⊕ V_{r}^{⊗N} ⊕ ⋯

as a Lie algebra via the usual bracket for associative algebras.

The complex free Lie algebra of rank r is the Lie subalgebra L(r) of T (V_{r}) generated by V_{r}. L(r) inherits the grading from T (V_{r}), so that

L(r) = H_{1}(r) ⊕ H_{2}(r) ⊕ ⋯ ⊕ H_{N}(r) ⊕ ⋯  (1)

The complex free N -step nilpotent Lie algebra of rank r is the Lie algebra

L(N, r) = \frac{L(r)}{∑_{i≥N+1} H_{i}(r)} .

By an abuse of notation we write this algebra as

L(N, r) = H_{1}(r) ⊕ H_{2}(r) ⊕ ⋯ ⊕ H_{N}(r)

where the bracket of two homogeneous elements of degree i and j is 0 if i + j > N. The elements of L(r) are sometimes called Lie polynomials. Several characterizations can be found in [6]. The universal enveloping algebra of L(r) is the tensor algebra T (V_{r}).

The subspace H_{i}(r) in (1) is the subspace of homogeneous Lie polynomials of degree i, which is the sum of H_{i_{1},i_{2},...,i_{r}}(r), the subspaces of homogeneous Lie polynomials of multidegree (i_{1}, i_{2}, …, i_{r}), with i_{1} + ⋯ + i_{r} = i. If M_{r}(i) = dim H_{i}(r) and M_{r}(i_{1}, i_{2}, …, i_{r}) = dim H_{i_{1},i_{2},...,i_{r}}(r), then

M_{r}(i) = \frac{1}{i} ∑_{d|i} μ(d)r^{i/d}  (2)

M_{r}(i_{1}, i_{2}, …, i_{r}) = \frac{1}{i} ∑_{d|i_{k}} μ(d)\frac{(\frac{i}{d})!}{(\frac{i_{1}}{d} )! … (\frac{i_{r}}{d} )!}  (3)

Explicit basis can be constructed for H_{i}(r). One well known basis is the Hall basis H(i), which is recursively defined (c.f. [7]); H(1) = X, H(2) is a subset of [H(1), H(1)] and in general H(i) is a subset of ∪_{j+k=i}[H(j), H(k)]. We take H = ∪_{k=1}^{N}H(k) as a basis for L(N, r).

Remark 1.1. All generators {X_{1}, …, X_{r}} appear the same number of times, if counted with multiplicity, in all the different Lie polynomials in H(i). In fact, the number of times that X_{k} appears in H(i) is

T_{k}(i) = ∑_{j=1}^{i−1} j ∑_{i_{1}+···+i_{k−1}+i_{k+1}+···+i_{r}=i−j} M(i_{1}, …, i_{k−1}, j, i_{k+1}, …, i_{r}).

It is straightforward to check, using (3), that T_{1}(i) = ⋯ = T_{r}(i). Since each element in H(i) is an i-bracket, it is clear that this common number is T (i) = \frac{M_{r}(i)i}{r}.

We include here some background for the representation theory of GL(r, ℂ), we fix some notation and make some conventions. We assume all representations to be of finite dimension. Proofs, as well as the general theory, can be found in any standard book, e.g. [1].

From now on, we fix a basis B = {v_{1}, …, v_{r}} of V_{r} and we denote by {E_{ij} : i, j = 1, …, r} the canonical basis of End(V_{r}). Let 𝔤𝔩(V_{r}) be the Lie algebra of GL(r, ℂ) and fix the triangular decomposition 𝔤𝔩(V_{r}) = 𝔫^{−} ⊕ 𝔥 ⊕ 𝔫^{+}, where 𝔫^{−}, 𝔥 and 𝔫^{+} are the subalgebras consisting of endomorphisms whose matrices in the basis B are respectively strictly lower-triangular, diagonal and strictly uppertriangular. Now {E_{11}, …, E_{rr}} is a basis of the Cartan subalgebra 𝔥. We will denote the corresponding dual basis by {ϵ_{1}, …, ϵ_{r}}. In particular {ϵ_{i} − ϵ_{j} : i < j} is the set of positive roots corresponding to the triangular decomposition chosen above.

A linear functional λ on 𝔥 is called a weight if it takes integer values on the vectors E_{ii} − E_{jj} for all i < j. A weight is said to be a dominant weight if λ(E_{ii} − E_{jj}) ≥ 0. A partition of length r is an r-tuple of non-negative integers λ = (λ_{1}, …, λ_{r}) such that λ_{1} ≥ ⋯ ≥ λ_{r}. Any partition of length r defines a dominant weight, which we will denote again by λ, given by ∑ λ_{i}ϵ_{i}.

By a polynomial representation we mean a finite dimensional representation of GL(r, ℂ) such that the matrix entries are given by polynomial functions. It is well known that every polynomial representation of GL(r, ℂ) can be decomposed as a sum of irreducible polynomial subrepresentations. In each irreducible polynomial representation W there is a unique (up to scalars) non-zero vector v such that 𝔫^{+}.v = 0 and H.v = λ(H)v where λ is a dominant weight. Such a vector is called a highest weight vector of weight λ, and W is called an irreducible representation of highest weight λ. In addition, W = U(𝔫^{−}).v, where U(𝔫^{−}) is the enveloping algebra of 𝔫^{−}.

The isomorphism classes of irreducible polynomial representations of GL(V) are in one-to-one correspondence with the partitions of length r. Given λ, W_{λ} will denote an irreducible representation of highest weight λ.

A partition λ = (λ_{1}, …, λ_{r}) is often represented by its Young diagram Y (λ), a graphical arrangement of λ_{i} boxes in the i-th row starting in the first column. So, any polynomial representation of GL(r, ℂ) can be described (up to isomorphism) as a direct sum of Young diagrams with at most r rows. For example, to the partition λ = (4, 2, 1, 0) it corresponds the Young diagram

.

2. General results

The group GL(r, ℂ) acts naturally on V_{r} and thus acts on T (V_{r}), acting in each coordinate. This action is polynomial. Since

g[x, y] = g(x ⊗ y − y ⊗ x) = gx ⊗ gy − gy ⊗ gx = [gx, gy],

for any x, y ∈ T (V_{r}) and g ∈ GL(r, ℂ), it follows that GL(r, ℂ) acts on L(r) preserving each one of the components in (1).

More generally, if W is a GL(r, ℂ)-representation, then the exterior powers of W, Λ^{p}W, are also polynomial GL(r, ℂ)-representations with GL(r, ℂ) acting in each coordinate. In this case the induced action of the Lie algebra 𝔤𝔩(r, ℂ) on the exterior powers of W is given by

A(w_{1} ∧ … ∧ w_{p}) = ∑_{i=1}^{p} w_{1} ∧ … ∧ Aw_{i} ∧ … ∧ w_{p},

for A ∈ 𝔤𝔩(r, ℂ) and w_{i} ∈ W.

Recall that the homology of a Lie algebra 𝔤, with trivial coefficients, is the homology of the complex

\xymatrix{⋯ \ar[r] & Λ^{p}𝔤 \ar[r]^{∂_{p}} & Λ^{p−1}𝔤 \ar[r] & ⋯ \ar[r] & Λ^{2}𝔤 \ar[r]^{∂_{2}} & 𝔤 \ar[r] & ℂ \ar[r] & 0}  (4)

where

∂_{p}(x_{1} ∧ … ∧ x_{p}) = ∑_{i<j}(−1)^{i+j+1}[x_{i}, x_{j}] ∧ x_{1} ∧ … x̂_{i} … x̂_{j} … ∧ x_{p}.  (5)

The exterior algebra complex, that computes the homology of L(N, r), is a GL(r, ℂ)-module and it is easy to verify that the differential is a GL(r, ℂ)- morphism. Therefore, the homology groups H_{p}(𝔤) = ker ∂_{p}/ Im ∂_{p+1} inherit that GL(r, ℂ)-module structure.

2.1. Poincaré duality. All complex finite dimensional nilpotent Lie algebras 𝔤 enjoy Poincaré duality, that is dim H_{i}(𝔤) = dim H_{dim 𝔤−i}(𝔤).

We prove here that for L(N, r) a stronger version of this duality holds. It is not true that H_{i}(L(N, r)) and H_{n−i}(L(N, r)) (n = dim L(N, r)) are GL(r, ℂ)- isomorphic, but there is a simple formula relating these GL(r, ℂ)-modules.

Theorem 2.1. Let L(N, r) be the complex free N -step nilpotent Lie algebra of rank r and let n be its dimension. Let M_{r}(i) be the dimension of the i-th homogeneous component of L(N, r) and set T = ∑_{i=1}^{N} \frac{M_{r}(i)i}{r}. Then

H_{n−i}(L(N, r)) ≃ H_{i}(L(N, r))^{∗} ⊗ det^{T}

as GL(r, ℂ)-modules.

To prove this theorem we need some propositions.

From now on we set 𝔫 = L(N, r), n = dim L(N, r) and we fix a Hall basis for 𝔫, {e_{1}, …, e_{n}}.

Consider the bilinear form B on Λ𝔫 defined on homogeneous elements v ∈ Λ^{p}𝔫 and w ∈ Λ^{q}𝔫 by

B(v, w) = \begin{cases}0, & if p + q ≠ n; \\ λ, & if p + q = n; \end{cases}

here in the second case λ is such that v ∧ w = λ(e_{1} ∧ … ∧ e_{n}). Let us take {v_{i_{1}...i_{p}} = e_{i_{1}} ∧…∧e_{i_{p}} : i_{1} < ⋯ < i_{p}, p = 1 … n}∪{v_{Ø} = 1} as a basis of Λ𝔫. The Hodge map ∗ : Λ𝔫 ⟶ Λ𝔫 is defined by ∗v_{i_{1}...i_{p}} = cv\overline{i_{1}...i_{p}} (c = ±1), where \overline{i_{1} … i_{p}} is the ordered complement of \{i_{1}, …, i_{p}\} in \{1, …, n\} and v_{i_{1}...i_{p}\overline{i_{1}...i_{p}}} = cv_{1...n}. Since v ∧ ∗v = e_{1} ∧ … ∧ e_{n}, B(v, ∗v) = 1 and B is non-degenerate.

Lemma 2.2. If v, w ∈ Λ𝔫, then B(gv, gw) = (det g)^{T} B(v, w) for any g ∈ GL(r, ℂ).

Proof. We can assume that v ∈ Λ^{p}𝔫 and w ∈ Λ^{q}𝔫 with p + q = n. Let z be a generator of Λ^{n}𝔫 and let g = SU be the (multiplicative) Jordan decomposition of g; there exist h ∈ GL(r, ℂ) such that P = hUh^{−1} is upper-triangular with all diagonal entries equal to 1. Since P z = z, then also Uz = z. On the other hand, because of Remark 1.1

Sz = [\begin{matrix}λ_{1} \\ & . & . & . \\ & & & & λ_{r} \end{matrix}] z = (λ_{1}^{T} … λ_{r}^{T})z.

Therefore gz = (det g)^{T} z and B(gv, gw) = gv ∧ gw = g(v ∧ w) = g(λz) = λ(det g)^{T} z = (det g)^{T} (v ∧ w) = (det g)^{T} B(v, w).

Proposition 2.3. Λ^{n−i}𝔫 ≃ (Λ^{i}𝔫)^{∗} ⊗ det^{T} as GL(r, ℂ)-modules.

Proof. For each v ∈ Λ^{n−i}𝔫 let α_{v} ∈ (Λ^{i}𝔫)^{∗} be defined by α_{v}(w) = B(v, w). Let Ψ : Λ^{n−i}𝔫 ⟶ (Λ^{i}𝔫)^{∗} ⊗ ℂ be defined by Ψ(v) = α_{v} ⊗ 1, where ℂ is the GL(r, ℂ)- module det^{T}. Since B is non-degenerate, Ψ is a ℂ-isomorphism. Moreover, Ψ is a GL(r, ℂ)-isomorphism. In fact,

Ψ(gv)(w) = (α_{gv} ⊗ 1)(w) = B(gv, w) = B(gv, gg^{−1}w)
= (det g)^{T} B(v, g^{−1}w)

and

gΨ(v)(w) = g(α_{v} ⊗ 1)(w) = (gα_{v} ⊗ (det g)^{T})(w)
= (det g)^{T} B(v, g^{−1}w).

Lemma 2.4. The square

\xymatrix{(Λ^{n−i}𝔫)^{∗} ⊗ ℂ & (Λ^{n−(i+1)}𝔫)^{∗} ⊗ ℂ \ar[l]_{−(∂^{n−i})^{t}⊗id} \\ Λ^{i}𝔫 \ar[u]^{Ψ_{i}} & Λ^{i+1}𝔫 \ar[l]^{∂^{i+1}} \ar[u]_{Ψ_{i+1}}}

is commutative if i is even and anti-commutative if i is odd.

Proof. Take v ∈ Λ^{i+1}𝔫 and w ∈ Λ^{n−i}𝔫; we may assume that v = a_{j_{1}} ∧…∧a_{j_{i+1}} and w = b_{l_{1}} ∧ … ∧ b_{l_{n−i}} with a_{j_{p}}, b_{l_{q}} ∈ {e_{1}, …, e_{n}}. To make formulas easier to read we will use the following notations

a(j_{p}, j_{q}) = a_{j_{1}} ∧ … ∧ â_{j_{p}} ∧ … ∧ â_{j_{q}} ∧ … ∧ a_{j_{i+1}};
b(l_{r}, l_{s}) = b_{l_{1}} ∧ … ∧ b̂_{l_{r}} ∧ … ∧ b̂_{l_{s}} ∧ … ∧ b_{l_{n−i}}.

By the definition of ∂ and Ψ we have

∂^{i+1}v = ∑_{p<q}(−1)^{p+q+1}[a_{j_{p}}, a_{j_{q}}] ∧ a(j_{p}, j_{q});
Ψ_{i}(∂^{i+1}v)w = ∑_{p<q}(−1)^{p+q+1}[a_{j_{p}}, a_{j_{q}}] ∧ a(j_{p}, j_{q}) ∧ b_{l_{1}} ∧ … ∧ b_{l_{n−i}}.  (6)

On the other hand

−(∂^{n−i})^{t}(Ψ_{i+1}v)w = −(Ψ_{i+1}v)(∂^{n−i}w) = −v ∧ ∂^{n−i}w
= ∑_{r<s}(−1)^{r+s}a_{j_{1}} ∧ … ∧ a_{j_{i+1}} ∧ [b_{l_{r}}, b_{l_{s}}] ∧ b(l_{r}, l_{s}). (7)

We can identify v and w with the sets {a_{j_{1}}, …, a_{j_{i+1}}} and {b_{l_{1}}, …, b_{l_{n−i}}} respectively. Thus, since #v = i + 1 and #w = n − i it follows that #(v ∩ w) ≥ 1.

If #(v ∩ w) ≥ 3 then the sums in (6) and in (7) are both equal to 0. If #(v ∩ w) = 2 we may assume that a_{j_{po}} = b_{l_{ro}} = x and a_{j_{qo}} = b_{l_{so}} = y.

Hence,

Ψ_{i}(∂^{i+1}v)w = (−1)^{p_{0}+q_{0}+1}[x, y] ∧ a(j_{p_{0}}, j_{q_{0}}) ∧ b_{l_{1}} ∧ … ∧ b_{l_{n−i}}

= (−1)^{p_{0}+q_{0}+r_{0}+s_{0}+2i−4}[x, y] ∧ b_{l_{r_{0}}} ∧ b_{l_{s_{0}}} ∧ a(j_{p_{0}}, j_{q_{0}}) ∧ b(l_{r_{0}}, l_{s_{0}})

− (∂^{n−i})^{t}(Ψ_{i+1}v)w = (−1)^{r_{0}+s_{0}}a_{j_{1}} ∧ … ∧ a_{j_{i+1}} ∧ [x, y] ∧ b(l_{r_{0}}, l_{s_{0}})

= (−1)^{r_{0}+s_{0}+p_{0}+q_{0}+i−2}[x, y] ∧ a_{j_{p_{0}}} ∧ a_{j_{q_{0}}} ∧ a(j_{p_{0}}, j_{q_{0}}) ∧ b(l_{r_{0}}, l_{q_{0}})

If #(v ∩ w) = 1 we may assume that a_{j_{p_{0}}} = b_{l_{r_{0}}} = x. It turns out that those summands in (6) with p ≠ p_{0} and q ≠ p_{0} are zero. Suppose p = p_{0} in (6), then

Ψ_{i}(∂^{i+1}v)w = ∑_{p_{0}<q}(−1)^{p_{0}+q+1} \underbrace{[a_{j_{p_{0}}}, a_{j_{q}}] ∧ a(j_{p_{0}}, j_{q}) ∧ b_{l_{1}} ∧ … ∧ b_{l_{n−i}}}_{=c_{q}}

But c_{q} ≠ 0 if and only if the coefficient of a_{j_{q}} in the expansion of [a_{j_{po}}, a_{j_{q}}], as an element in 𝔫, is ≠ 0. Since the homogeneous degree of [a_{j_{po}}, a_{j_{q}}] is greater than that of a_{j_{p}} we conclude that c_{q} = 0 for all q and then the sum in (6) is equal to 0. The case when q = p_{0} follows similarly.

In an analogous way we conclude that the sum in (7) is equal to 0.

Proof. [Proof of Theorem 2.1] The homology of 𝔫 is the homology of the complex

\xymatrix{⋯ \ar[r] & Λ^{i+1}𝔫 \ar[r]^{∂^{i+1}} & Λ^{i}𝔫 \ar[r]^{∂^{i}} & Λ^{i−1}𝔫 \ar[r] & ⋯}

and its cohomology is the homology of the complex

\xymatrix{⋯ & (Λ^{i+1}𝔫)^{∗} \ar[l] & (Λ^{i}𝔫)^{∗} \ar[l]_{−(∂^{i+1})^{t}} & (Λ^{i−1}𝔫)^{∗} \ar[l]_{−(∂^{i})^{t}} & ⋯ \ar[l]}

Consider the following diagram of GL(r, ℂ)-modules and GL(r, ℂ)-morphisms with commutative or anti-commutative squares.

\xymatrix{⋯ (Λ^{i+1}𝔫)^{∗} ⊗ ℂ & (Λ^{i}𝔫)^{∗} ⊗ ℂ \ar[l]_{−(∂^{i+1})^{t}⊗id} & (Λ^{i−1}𝔫)^{∗} ⊗ ℂ ⋯ \ar[l]_{−(∂^{i})^{t}⊗id} \\ ⋯ Λ^{n−(i+1)}L \ar[u]_{Ψ} & Λ^{n−i}𝔫 \ar[l]_{∂^{n−i}} \ar[u]_{Ψ} & Λ^{n−(i−1)}𝔫 ⋯ \ar[l]_{∂^{n−(i−1)}} \ar[u]_{Ψ}}

Since the upper complex computes the cohomology of 𝔫 it follows that Ψ induces a GL(r, ℂ)-isomorphism

H_{n−i}(𝔫) ≃ H^{i}(𝔫) ⊗ det^{T} .

On the other hand the linear isomorphism α : H^{i}(𝔫) ⟶ H_{i}(𝔫)^{∗} defined by α([f]) = f_{∗} for any [f] ∈ H^{i}(𝔫), where f_{∗}([z]) = [f(z)] (recall that Λ^{i}𝔫^{∗} ≃ (Λ^{i}𝔫)^{∗}), is a GL(r, ℂ)-morphism. So we arrive to the desired GL(r, ℂ)-isomorphism

H_{n−i}(𝔫) ≃ H_{i}(𝔫)^{∗} ⊗ det^{T}

Remark 2.5. It follows that as GL(r, ℂ)-modules H_{n} ≃ det^{T}, for every r ≥ 2.

Remark 2.6. The previous isomorphism can be interpreted in terms of Young diagrams as follows.

For each diagram Y_{λ} that fits in the r × T rectangle let Y_{λ}^{c} be the diagram obtained by rotating 180 degrees the complementary arrangement of Y_{λ} in the r × T rectangle.

For example, if r = 4 we have T = 6, the diagram of the representation

det^{T} is the 4×6 rectangle  and for Y_{λ} =  the corresponding diagram

Y_{λ}^{c} = .

Now, Y_{λ} ∈ H_{n−i} if and only if Y_{λ}^{c} ∈ H_{i}. This follows from the fact that λ is a highest weight of a representation W of GL(r, ℂ) if and only if (T, …, T) − λ is a highest weight of W^{∗} ⊗ det^{T}.

2.2. Homology stabilization. The inclusion {x_{1}, …, x_{r}} ↪ {x_{1}, …, x_{r}, x_{r+1}} induces a ℂ-monomorphism T (V_{r}) ↪ T (V_{r+1}), a Lie monomorphism L(N, r) ↪ L(N, r + 1) and a ℂ-monomorphism Λ^{p}L(N, r) ↪ Λ^{p}L(N, r + 1) for each 0 ≤ p ≤ n. Moreover, by considering GL(r, ℂ) as the subgroup of GL(r + 1, ℂ) consisting of matrices ( \begin{matrix}A & 0 \\ 0 & 1 \end{matrix} ), with A ∈ GL(r, ℂ), all these maps are GL(r, ℂ)-morphisms.

Let us denote 𝔫′ = L(N, r + 1) and consider the homology groups H_{p}(𝔫) and H_{p}(𝔫′), for 0 ≤ p ≤ n, as GL(r, ℂ) and GL(r + 1, ℂ)-modules respectively.

Lemma 2.7. If 0 ≠ [v] ∈ H_{p}(𝔫) is a highest weight vector, then 0 ≠ [v] ∈ H_{p}(𝔫′) is also a highest weight vector.

Proof. Let λ = (λ_{1}, …, λ_{r}) be the weight of v ∈ ker ∂_{p}. It is clear that v ∈ ker ∂′_{p} and since E_{r+1,r+1}v = 0, then λ′ = (λ_{1}, …, λ_{r}, 0) is the weight of v ∈ ker ∂′_{p}. In addition E_{ij}v = 0 for 1 ≤ i < j = r + 1 and therefore v is a highest weight vector in ker ∂′_{p}.

Suppose now that v = ∂′_{p+1}(w) for some w ∈ Λ^{p+1}𝔫′. We may assume that w is a weight vector since w = weight vector + cycle. Moreover the weight of w is λ′, because ∂′_{p+1} is a GL(r + 1, ℂ)-morphism, and thus w ∈ Λ^{p+1}𝔫. Being ∂_{p+1} = ∂′_{p+1}|_{Λ^{p+1}𝔫}, it follows that v = ∂_{p+1}(w) or 0 = [v] ∈ H_{p}(𝔫).

Corollary 2.8. Each Young diagram, counted with multiplicity, in the decomposition of H_{p}(𝔫) is in the decomposition of H_{p}(𝔫′).

Proof. Let Y_{λ} be a Young diagram in the decomposition of H_{p}(𝔫) and take v a highest weight vector of weight λ in H_{p}(𝔫). By the previous Lemma v is a highest weight vector in H_{p}(𝔫′) of weight λ′ + (λ_{1}, …, λ_{r}, 0), so that the corresponding Young diagram is Y_{λ}.

Theorem 2.9. If r ≥ pN, then the Young diagram decompositions of the homology groups H_{p}(L(N, r)) and H_{p}(L(N, r + 1)) are identical.

Proof. Let Y_{λ′} be a Young diagram in the decomposition of H_{p}(L(N, r + 1)) and take v a corresponding highest weight vector with weight λ′ = (λ_{1}, …, λ_{r+1}). Since v ∈ Λ^{p}L(N, r+), then λ_{1} + ⋯ + λ_{r+1} ≤ pN and therefore λ_{r+1} = 0; this means that v ∈ Λ^{p}L(N, r) and finally Y_{λ′} = Y_{λ}, with λ = (λ_{1}, …, λ_{r}) and Y_{λ} is in the Young decomposition of H_{p}(L(N, r)).

Remark 2.10. The Young diagram decomposition of H_{p}(L(N, s)), if s ≤ r, can be read directly from that of H_{p}(L(N, r)). In fact, H_{p}(L(N, s)) decomposes as the sum of all those Young diagrams in H_{p}(L(N, r)) with at most s rows.

Remark 2.11. Theorem 2.9 says that in order to compute H_{p}(L(N, r)) for all r’s it is enough to do it for r = pN. It turns out, in the cases treated here, that a smaller r is enough as well.

2.3. Minimal weights. Let 0 → 𝔥 → L → 𝔤 → 0 be a free presentation of 𝔤, an arbitrary Lie algebra over ℂ. Let F be the augmentation ideal of U(L) (universal enveloping algebra of L) and R be the ideal of U(L) generated by i(𝔥), where i : L → U(L) is the canonical map. We have then a Gruenberg like formula for the homology of 𝔤.

Theorem 2.12. Let 𝔤, F and R be as above. Then

H_{2n+1}(𝔤) = \frac{FR^{n} ∩ R^{n}F}{FR^{n}F + R^{n+1}} ,
H_{2n}(𝔤) = \frac{R^{n} ∩ FR^{n−1}F}{R^{n}F + FR^{n}} .

These formulas for Lie algebra homology are the analogous of those for group homology given by Gruenberg in [2] and [3]. Since U(L) is a tensor algebra the ideals F and R are free as left ideals. The proof is then almost identical as Gruenberg’s proof.

Definition 2.13. Given a Young diagram Y_{λ} we define its total weight as |Y_{λ}| = λ_{1} + ⋯ + λ_{r}, if λ = (λ_{1}, …, λ_{r}).

Theorem 2.14. Let 𝔫 be a complex free N -step nilpotent Lie algebra of any rank. If Y_{λ} is a Young diagram in the decomposition of H_{i}(𝔫), then

nN + (n + 1) ≤ |Y_{λ}|, if i = 2n + 1
nN + n ≤ |Y_{λ}|, if i = 2n

Proof. In all cases U(L) is a tensor algebra, F is the ideal of polynomials without constant term and R is the ideal generated by the Lie polynomials of degree ≥ N + 1.

3. Explicit computations

To compute explicitly the homology of L(N, r) we can proceed directly with the complex (4). The first step is the decomposition of the exterior powers of L(N, r) as a sum of irreducible GL(r, ℂ)-representations. Then we compute the differential ∂ (see 5) on the highest weight vectors corresponding to the previous decomposition to finally determine the homology groups.

On the other hand we can use a Laplacian Δ, as in [5]. Recall that two linear operators ∂ and d on Λ𝔫 such that ∂^{2} = d^{2} = 0 are disjoint if

1. d∂(x) = 0 ⟹ ∂(x) = 0;

2. ∂d(x) = 0 ⟹ d(x) = 0.

In this case there is a canonical isomorphism from the kernel of the Laplacian Δ = d∂ + ∂d to the derived space of homology of Λ𝔫, \frac{ker d}{Im d}. If ∂ = d^{∗}, the adjoint of d with respect to an inner product defined on Λ𝔫, then d and ∂ are disjoint.

Proposition 3.1. There is a GL(r, ℂ)-morphism d in Λ𝔫 which is disjoint to ∂.

Proof. Consider the unitary group U(r) ⊆ GL(r, ℂ) and 𝔲(r) its Lie algebra. There is on 𝔫 a U(r)-invariant inner product. (It is unique, up to scalars, in each irreducible component of 𝔫.) We extend this inner product to the exterior powers of 𝔫, via the determinant, remaining U(r)-invariant. Let d = ∂^{∗} be the adjoint of ∂. d is U(r)-equivariant and therefore also 𝔲(r)-equivariant. Since d is ℂ-linear and 𝔤𝔩(r, ℂ) is the complexification of 𝔲(r) it turns out to be a 𝔤𝔩(r, ℂ)-morphism and then a GL(r, ℂ)-morphism.

Let d_{p} = d|_{Λ^{p}𝔫} and let {f_{1}, …, f_{n}} be an orthonormal basis of 𝔫. Then

d_{p}(x_{1} ∧ … ∧ x_{p}) = ∑_{i=1}^{p}(−1)^{i+1}d_{1}(x_{i}) ∧ x_{1} ∧ … x̂_{i} … ∧ x_{p},

where d_{1}(f_{k}) = ∑_{i<j} c_{ij}^{k}f_{i} ∧ f_{j} and [f_{i}, f_{j}] = ∑ c_{ij}^{k}f_{k}. It is straightforward to verify the equation ⟨∂_{p+1}v, w⟩ = ⟨v, d_{p}w⟩ on the induced orthonormal basis of Λ𝔫 (recall the definition of ∂_{p} in (4)).

We used both methods in the cases of rank 2 algebras and L(III, 3) and the direct computations in the case of L(III, r).

We may notice that some of the spaces involved are very big. For example, the algebra L(III, 7) is of dimension 140, hence its fourth exterior power has dimension 15329615 and its fifth exterior power has dimension 416965528. These spaces are involved in the computation of H_{4}(L(III, 7)).

3.1. Presentation of data. In all cases the homology groups are displayed as lists of Young diagrams, that is as sums of irreducible GL(r, ℂ)-representations. The dimension of each group and the total dimension are also given. We notice that in all cases H_{0} = ℂ, so we omit it. For large algebras we give the corresponding number T and therefore, by virtue of Theorem 2.1 and Remark 2.6, we only show half of the homology groups. We do not present the highest weight cycles corresponding to each Young diagram because of the length of all this data. However, we can make them available to the interested reader.

For the cases were a Laplacian has been used we include more information. We write down Hall basis for these algebras and we give the U(r)-invariant inner products and the corresponding d_{1} operators.

3.2. Rank 2 algebras. Hall basis for the first 5 homogeneous components of the free Lie algebra L(2) are listed below.

H_{1} = ⟨x, y⟩
= ⟨r_{1}, r_{2}⟩
H_{2} = ⟨[xy]⟩
= ⟨t_{1}⟩
H_{3} = ⟨[x[xy]], [y[xy]]⟩
= ⟨u_{1}, u_{2}⟩
H_{4} = ⟨[x[x[xy]]], [y[x[xy]]], [y[y[xy]]]⟩
= ⟨v_{1}, v_{2}, v_{3}⟩
H_{5} = ⟨[x[x[x[xy]]]], [y[x[x[xy]]]], [y[y[x[xy]]]], [y[y[y[xy]]]], [[xy][x[xy]]],
[[xy][y[xy]]]⟩
= ⟨w_{1}, w_{2}, w_{3}, w_{4}, w_{5}, w_{6}⟩

All the basis vectors are weight vectors. Since weight vectors of different weights are orthogonal, we only compute the norms and the non-zero products.

||r_{1}|| = 1, ||r_{2}|| = 1;

||t_{1}|| = 1;

||u_{1}|| = 1, ||u_{2}|| = 1;

||v_{1}|| = 1, ||v_{2}|| = \frac{1}{2}, ||v_{3}|| = 1;

||w_{1}|| = 1, ||w_{2}|| = \frac{7}{9}, ||w_{3}|| = \frac{4}{9}, ||w_{4}|| = 1, ||w_{5}|| = 1, ||w_{6}|| = 1,

< w_{2}, w_{5} >= −\frac{2}{3}, < w_{3}, w_{6} >= −\frac{1}{3}.

This inner product allows us to compute d_{1} (and hence d). We have,

d_{1}(r_{1}) = 0, d_{1} = (r_{2}) = 0;

d_{1}(t_{1}) = r_{1} ∧ r_{2};
d_{1}(u_{1}) = r_{1} ∧ t_{1}, d_{1}(u_{2}) = r_{2} ∧ t_{1};

d_{1}(v_{1}) = r_{1} ∧ u_{1}, d_{1}(v_{2}) = \frac{1}{2} r_{1} ∧ u_{2} + \frac{1}{2} r_{2} ∧ u_{1}, d_{1}(v_{3}) = r_{2} ∧ u_{2}; d_{1}(w_{1}) = r_{1} ∧ v_{1}, d_{1}(w_{2}) = \frac{2}{9} r_{1} ∧ v_{2} + \frac{7}{9} r_{2} ∧ v_{1} − \frac{2}{3} t_{1} ∧ u_{1}, d_{1}(w_{3}) = \frac{1}{9} r_{1} ∧ v_{3} + \frac{8}{9} r_{2} ∧ v_{2} − \frac{1}{3} t_{1} ∧ u_{2}, d_{1}(w_{4}) = r_{2} ∧ v_{3};

d_{1}(w_{5}) = \frac{2}{3} r_{1} ∧ v_{2} + t_{1} ∧ u_{1} − \frac{2}{3} r_{2} ∧ v_{1},

d_{1}(w_{6}) = \frac{2}{3} r_{1} ∧ v_{3} + t_{1} ∧ u_{2} − \frac{2}{3} r_{2} ∧ v_{2}.

3.3. The homology of L(III, 2).

Group Young decomposition Dimension
H_{1}  2
H_{2}  3
H_{3}  3
H_{4}  2
H_{5}  1

Total homology: 12 3.4. The homology of L(IV, 2).

Group  Young decomposition  Dimension
H_{1}  2
H_{2}  6
H_{3}  13
H_{4}  16
H_{5}  13
H_{6}  6
H_{7}  2
H_{8}  1

Total homology: 60 3.5. The homology of L(V, 2).

320 Tirao

Group Young decomposition

Dim

H_{1}  2
H_{2}  9
H_{3}  39
H_{4}  85
H_{5}  145

H_{6}  206

H_{7}  258

det^{T} = (T = 26) . Total homology: 1232

3.6. The homology of L(III, 3). A Hall basis for L(III, 3) is given by

H_{1} = ⟨x, y, z⟩
= ⟨r_{1}, r_{2}, r_{3}⟩
H_{2} = ⟨[xy], [xz], [yz]⟩
= ⟨t_{1}, t_{2}, t_{3}⟩
H_{3} = ⟨[x[xy]], [x[xz]], [y[xy]], [y[xz]], [y[yz]], [z[xy]], [z[xz]], [z[yz]]⟩
= ⟨u_{1}, u_{2}, …, u_{8}⟩

As in the case of rank 2 algebras all basis vectors are weight vectors and therefore we only compute the norms and the non-zero products (vectors of different weights are orthogonal).

||r_{1}|| = 1, ||r_{2}|| = 1, ||r_{3}|| = 1;
||t_{1}|| = 1, ||t_{2}|| = 1, ||t_{3}|| = 1;
||u_{1}|| = 1, ||u_{2}|| = 1, ||u_{3}|| = 1, ||u_{5}|| = 1, ||u_{7}|| = 1, ||u_{8}|| = 1,

||u_{4}|| = \frac{2}{3}, ||u_{6}|| = \frac{2}{3};

< u_{4}, u_{6} >= \frac{1}{3}.

This inner product allow us to compute d_{1} (and hence d). We have,

d_{1}(r_{1}) = 0, d_{1} = (r_{2}) = 0, d_{1}(r_{3});
d_{1}(t_{1}) = r_{1} ∧ r_{2}, d_{1}(t_{2}) = r_{1} ∧ r_{3}, d_{1}(t_{3}) = r_{2} ∧ r_{3};
d_{1}(u_{1}) = r_{1} ∧ t_{1}, d_{1}(u_{2}) = r_{1} ∧ t_{2}, d_{1}(u_{3}) = r_{2} ∧ t_{1};
d_{1}(u_{5}) = r_{2} ∧ t_{3}, d_{1}(u_{7}) = r_{3} ∧ t_{2}, d_{1}(u_{8}) = r_{3} ∧ t_{3};
d_{1}(u_{4}) = \frac{1}{3} r_{1} ∧ t_{3} + \frac{2}{3} r_{2} ∧ t_{2} + \frac{1}{3} r_{3} ∧ t_{1},
d_{1}(u_{6}) = −\frac{1}{3} r_{1} ∧ t_{3} + \frac{1}{3} r_{2} ∧ t_{2} + \frac{2}{3} r_{3} ∧ t_{1} .

Group Young decomposition

Dim

H_{1}  3
H_{2}  18
H_{3}  70
H_{4}  171
H_{5}  327

H_{6}  462

H_{7}  504

det^{T} = (T = 11) . Total homology: 2608

3.7. The first homology groups of L(III, r).

Group Young decomposition

Dim

H_{1}  p_{1}(r)
H_{2}  p_{2}(r)
H_{3}  p_{3}(r)

H_{4}  p_{4}(r)

The polynomials p_{i}(r) are:

p_{1}(r) = r,

p_{2}(r) = \frac{1}{4}r^{2}(r + 1)(r − 1) ∼ \frac{1}{4}r^{4} ,

p_{3}(r) = \frac{1}{420}(10r^{4} + 14r^{3} + 31r^{2} − 56r − 74)r(r + 1)(r − 1) ∼ \frac{1}{42}r^{7} ,

p_{4}(r) = \frac{1}{1209600}(1001r^{7} + 3975r^{6} + 18371r^{5} − 4635r^{4} − 54436r^{3}

−100500r^{2} + 111744r + 120960)r(r + 1)(r − 1) ∼ \frac{1001}{1209600}r^{10} .

Aknowledgments This paper was written while visiting the Heinrich-Heine-Universität in Düsseldorf. My deepest gratitude to the Matematisches Institut for their hospitality and particular thanks to Professor Fritz Grunewald for his support and encouragment.

References

[1] Fulton, W., and J. Harris, “Representation Theory: A First Course, ” Graduate Texts in Math. 129, Springer-Verlag, New York, 1991.

[2] Gruenberg, K., Resolutions by relations, J. London Math. Soc. 35 (1960), 481–494.

[3] —, The universal coefficient theorem in the cohomology of groups, J. London Math. Soc. 43 (1968), 239–241.

[4] Grassberger, J., A. King and P. Tirao, On the homology of free 2-step nilpotent Lie algebras, ICTP Preprint No. IC/99/14.

[5] Kostant, B., Lie algebra cohomology and the generalized Borel-Weil theorem, Ann. of Math. 74 (1961), 329–387.

[6] Reutenauer, C., “Free Lie Algebras,” London Mathematical Society Monographs. New Series.7. Oxford, Clarendon Press, 1993.

[7] Serre, J. P., “Lie Algebras and Lie Groups,” Lecture Notes in Mathematics 1500, Springer-Verlag, 1992.

[8] Sigg, S., Laplacian and homology of free 2-step nilpotent Lie algebras, J. of Algebra 185 (1996), 144–161.

[9] Zhuravlev, V., A free Lie algebra as a module over the full linear group, Sbornik: Mathematics 187 (1996), 215–236.

Paulo Tirao CIEM-FaMAF Universidad Nacional de Córdoba Ciudad Universitaria, 5000 Córdoba, Argentina. ptirao@mate.uncor.edu

Received August 15, 2000 and in final form November 19, 2001