PUBLICATIONS DE L’INSTITUT MATHÉMATIQUE Nouvelle série, tome 94 (108) (2013), 99–109 DOI: 10.2298/PIM1308099D

C_{∞} -STRUCTURE ON THE COHOMOLOGY OF THE FREE 2-NILPOTENT LIE ALGEBRA

Michel Dubois-Violette and Todor Popov

Abstract. We consider the free 2-step nilpotent Lie algebra and its cohomology ring. The homotopy transfer induces a homotopy commutative algebra on its cohomology ring which we describe. We show that this cohomology is generated in degree 1 as C_{∞} -algebra only by the induced binary and ternary operations.

1. Homotopy algebras

The homotopy associative algebras, or A_{∞} -algebras were introduced by Jim Stasheff in the 1960’s as a tool in algebraic topology for studying ‘group-like’ spaces. Homotopy algebras received a new attention and further development in the 1990’s after the discovery of their relevance into a multitude of topics in algebraic geometry, symplectic and contact geometry, knot theory, moduli spaces and deformation theory.

Definition 1.1. (A_{∞} -algebra) A homotopy associative algebra, or A_{∞} -algebra, over a field 𝕂 is a ℤ-graded vector space A = ⨁ _{i∈ℤ} A^{i} endowed with a family of graded mappings (operations) m_{n} : A^{⊗n} → A, deg(m_{n}) = 2 − n, n ⩾ 1 satisfying the Stasheff identities SI(n) for n ⩾ 1

SI(n) : ∑_{r+s+t=n} (−1)^{r+st} m_{r+1+t} (Id^{⊗r} ⊗m_{s} ⊗ Id^{⊗t} ) = 0 r ⩾ 0, t ⩾ 0, s ⩾ 1,

where the sum runs over all decompositions n = r + s + t. Throughout the text we assume the Koszul sign convention (f ⊗ g)(x ⊗ y) = (−1)^{|g||x|} f(x) ⊗ g(y).

A morphism of two A_{∞} -algebras A and B is a family of graded maps f_{n} : A^{⊗n} → B for n ⩾ 1 with degf_{n} = 1 − n such that the following conditions hold

∑_{r+s+t=n} (−1)^{r+st} f_{r+1+t} (Id^{⊗r} ⊗m_{s} ⊗ Id^{⊗t} ) = ∑_{1⩽r⩽n} (−1)^{S}m_{r}(f_{i_{1}} ⊗ f_{i_{2}} ⊗ ⋯⊗ f_{i_{r}} )

2010 Mathematics Subject Classification: Primary 17B35, 17B56; Secondary 18G10, 17D98.

Partially supported by Office of External Activities of ICTP, Trieste and CDC of International Mathematical Union.

where the sum is over all decompositions i_{1} + ⋯+ i_{r} = n and the sign (−1)^{S} on the right-hand side is determined by

S = (r − 1)(i_{1} − 1) + (r − 2)(i_{2} − 1) + ⋯+ 2(i_{r−2} − 1) + (i_{r−1} − 1).

The morphism f is a quasi-isomorphism of A_{∞} -algebras if f_{1} is a quasi-isomorphism. It is strict if f_{i} = 0 for all i ≠ 1. The identity morphism of A is the strict morphism f such that f_{1} is the identity of A.

We define the shuffle product Sh_{p,q} : A^{⊗p} ⊗ A^{⊗q} → A^{⊗p+q} by

(a_{1} ⊗ ⋯⊗ a_{p}) � (a_{p+1} ⊗ ⋯⊗ a_{p+q} ) = ∑_{σ∈Sh_{p,q}} ± sgn(σ) a_{σ^{−1} (1)} ⊗ ⋯⊗ a_{σ^{−1} (p+q)}

where the sum runs over all (p, q)-shuffles Sh_{p,q}, i.e., over all permutations σ ∈ S_{p+q} such that σ(1) < σ(2) < ⋯< σ(p) and σ(p + 1) < σ(p + 2) < ⋯< σ(p + q) and the signs ± on the right-hand side are fixed from the cohomological degrees â_{i} of the elements a_{i} according to the place permutation action in the tensor powers of graded spaces.

Definition 1.2. (C_{∞} -algebra [10]) A homotopy commutative algebra, or C_{∞} - algebra, is an A_{∞} -algebra {A, m_{n}} such that each operation m_{n} vanishes on nontrivial shuffles m_{n}((a_{1} ⊗ ⋯⊗ a_{p}) � (a_{p+1} ⊗ ⋯⊗ a_{n})) = 0, 1 ⩽ p ⩽ n − 1.

In particular for m_{2} we have m_{2}(a ⊗ b − (−1)^{âb̂}b ⊗ a) = 0, so a C_{∞} -algebra such that m_{n} = 0 for n ⩾ 3 is a (super-)commutative DGA.

A morphism of C_{∞} -algebras is a morphism of A_{∞} -algebras vanishing on nontrivial shuffles f_{n}((a_{1} ⊗ ⋯⊗ a_{p}) � (a_{p+1} ⊗ ⋯⊗ a_{n})) = 0, 1 ⩽ p ⩽ n − 1.

2. Homotopy transfer theorem

Lemma 2.1. Every cochain complex (A, d) of vector spaces over a field 𝕂 has its cohomology H^{•}(A) as a deformation retract.

One can always choose a vector space decomposition of the cochain complex (A, d) such that A^{n} ≅ B^{n} ⊕ H^{n} ⊕ B^{n+1} where H^{n} is the cohomology and B^{n} is the space of coboundaries, B^{n} = dA^{n−1} . We choose a homotopy h : A^{n} → A^{n−1} which identifies B^{n} with its copy in A^{n−1} and is 0 on H^{n} ⊕B^{n+1} . The projection p to the

cohomology and the cocycle-choosing inclusion i given by \xymatrix{A^{n} \ar[r]^{p} & H^{n} \ar[l]^{i}} are chain

homomorphisms, satisfying the additional side conditions: hh = 0, hi = 0, ph = 0. With these choices done the complex (H^{•}(A), 0) is a deformation retract of (A, d)

[figure: h] \xymatrix{(A, d) \ar[r]^{p} & (H^{•}(A), 0) \ar[l]^{i}} , pi = Id_{H^{•} (A)} , ip − Id_{A} = dh + hd.

Let now (A, d, μ) be a DGA, i.e., A is endowed with an associative product μ compatible with d. The cochain complexes (A, d) and its contraction H^{•}(A) are homotopy equivalent, but the associative structure is not stable under homotopy equivalence. However the associative structure on A can be transferred to an A_{∞} -structure on a homotopy equivalent complex, a particular interesting complex being the deformation retract H^{•}(A). For a friendly introduction to homotopy transfer theorems in much broader context we refer the reader to the textbook [14, Chapter 9].

Theorem 2.1 (Kadeishvili [10]). Let (A, d, μ) be a (commutative) DGA over a field 𝕂. There exists an A_{∞} -algebra (C_{∞} -algebra) structure on the cohomology H^{•}(A) and an A_{∞} (C_{∞} )-quasi-isomorphism

f_{k} : (⊗ ^{k}H^{•}(A), {m_{j} }) → (A, {d, μ, 0, 0, …})

such that the inclusion f_{1} = i : H^{•}(A) → A is a cocycle-choosing homomorphism of cochain complexes. The differential m_{1} on H^{•}(A) is zero (m_{1} = 0) and m_{2} is the strictly associative operation induced by the multiplication on A. The resulting structure is unique up to quasi-isomorphism.

Kontsevich and Soibelman [12] gave explicit expressions for the higher operations of the induced A_{∞} -structure as sums over decorated planar binary trees with one root where all leaves are decorated by the inclusion i, the root by the projection p, the vertices by the product μ of the (commutative) DGA (A, d, μ) and the internal edges by the homotopy h. The C_{∞} -structure implies additional symmetries on trees.

For instance the operation m_{2} of the induced A_{∞} -structure on H^{•}(A) looks like

[figure: i; i; m_{2}(x, y) := pμ(i(x), i(y)) or m_{2} =; μ; p]

and the ternary one m_{3}(x, y, z) = pμ(i(x), hμ(i(y), i(z))) − pμ(hμ(i(x), i(y)), i(z)) is the sum of two planar binary trees with three leaves

[figure: i; i; i; μ; h; m_{3} =; μ; p; −; i; i; i; μ; h; μ; p]

3. Homology and cohomology of a Lie algebra 𝔤

A non-minimal projective (in fact free) resolution of the trivial U𝔤-module 𝕂, C(𝔤) →^{ϵ} 𝕂 is given by the standard Chevalley–Eilenberg chain complex C_{•}(𝔤) = (U𝔤 ⊗_{𝕂} ∧^{p}𝔤, d_{p}) with differential maps

d_{p}(u ⊗ x_{1} ∧ ⋯∧ x_{p}) = ∑_{i} (−1)^{i+1} ux_{i} ⊗ x_{1} ∧ ⋯∧ x̂_{i} ∧ ⋯∧ x_{p}

+ ∑_{i<j} (−1)^{i+j} u ⊗ [x_{i}, x_{j}] ∧ x_{1} ∧ ⋯∧ x̂_{i} ∧ ⋯∧ x̂_{j} ∧ ⋯∧ x_{p}

The homologies H_{n}(𝔤, 𝕂) of the Lie algebra 𝔤 with trivial coefficients are given by the homologies of the derived complex 𝕂 ⊗_{U𝔤} C_{•}(𝔤)

Tor_{n}^{U𝔤}(𝕂, 𝕂) ≅ H_{n}(𝕂 ⊗_{U𝔤} C_{•}(𝔤)) = H_{n}(𝔤, 𝕂).

The complex 𝕂⊗_{U𝔤}C_{•}(𝔤) is the chain complex with degrees ⋀ ^{•} 𝔤 = 𝕂⊗_{U𝔤}U𝔤⊗⋀ ^{•} 𝔤 and differentials ∂_{p} := id ⊗_{U𝔤} d_{p} : ⋀ ^{p} 𝔤 → ⋀ ^{p−1} 𝔤 induced by the extension as coderivation of the Lie bracket ∂_{2} := −[ · , ·] : ⋀ ^{2} 𝔤 → 𝔤.

The dual cochain complex Hom_{U𝔤}(C(𝔤), 𝕂) = (⋀ ^{•} 𝔤^{∗}, δ) has coboundary map δ^{p} : ⋀ ^{p} 𝔤^{∗} → ⋀ ^{p+1} 𝔤^{∗} (being transposed to the differential ∂_{p+1} ) which is the extension as derivation of the dualization of the Lie bracket δ^{1} := [ · , ·]^{∗} : 𝔤^{∗} → ⋀ ^{2} 𝔤^{∗}. One calculates the cohomologies^{1} of the Lie algebra 𝔤 as

Ext_{U𝔤}^{n}(𝕂, 𝕂) ≅ H^{n}(Hom_{U𝔤}(C(𝔤), 𝕂)) = H^{n}(𝔤, 𝕂).

Hence the algebra (⋀ ^{•} 𝔤^{∗}, δ) equipped with δ is a (super)commutative DGA and the Yoneda algebra Ext_{U𝔤}^{•}(𝕂, 𝕂) = ⨁ _{n} Ext_{U𝔤}^{n}(𝕂, 𝕂) has the structure of commutative associative algebra. Moreover due to the Kadeishvili theorem the Yoneda algebra Ext_{U𝔤}^{•}(𝕂, 𝕂) = H^{•}(𝔤, 𝕂) is a C_{∞} -algebra which stems from the homotopy transfer of the wedge product ∧ on cohomology classes H^{i}(𝔤, 𝕂) ∧ H^{j} (𝔤, 𝕂) → H^{i+j} (𝔤, 𝕂).

4. Abelian Lie algebra 𝔥 = V

Let us take as a basic example the abelian Lie algebra 𝔥, that is, the free nilpotent Lie algebra of rank 1 generated by a finite dimensional vector space V. The Lie bracket of 𝔥 is trivial [V, V] = 0. The universal enveloping algebra of the abelian Lie algebra 𝔥 = V is the symmetric algebra U(𝔥) ≅ S(V). The Chevalley– Eilenberg complex C_{•}(𝔥) = S(V) ⊗_{𝕂} Λ^{•}V yields the resolution of the trivial U(𝔥)- module 𝕂

(4.1) 0 → S(V) ⊗ Λ^{dim V}V → S(V) ⊗ Λ^{dim V −1} V → …

⋯→ S(V) ⊗ Λ^{2}V → S(V) ⊗ V → S(V) → 𝕂 → 0.

The derived complex 𝕂 ⊗_{U𝔥} C(𝔥) has zero differential and the Chevalley–Eilenberg resolution turns out to be minimal (which is not the case in general)

H_{n}(𝔥, 𝕂) ≅ H_{n}(𝕂 ⊗_{U𝔥} C(𝔥)) ≅ Λ^{n}V.

The Chevalley–Eilenberg resolution coincides with the Koszul complex K(A) = A ⊗ (A^{!})^{∗} of the symmetric algebra A = S(V). The Koszul dual algebra of the symmetric algebra is the exterior algebra S(V)^{!} = ΛV^{∗}. A quadratic algebra is said to be a Koszul algebra when its Koszul complex K_{•}(A) = A ⊗ (A_{•}^{!})^{∗} is acyclic everywhere except in degree 0 (where its homology is 𝕂). Then the Koszul complex yields a minimal projective (in fact free) resolution by (left) A-modules of the trivial A-module 𝕂

1In the presence of any metric on a nilpotent Lie algebra 𝔤 one has δ := ∂^{∗}(see below).

K(A) →^{ϵ} 𝕂 → 0.

In particular the resolution (4.1) is the same as the the resolution by the Koszul complex K_{n}(S(V)) = S(V)⊗Λ^{n}V^{∗} thus the algebra S(V) is a Koszul algebra. One has an equivalent definition of Koszul algebra based on the following proposition.

Proposition 4.1. A finitely generated quadratic algebra A is Koszul iff its Yoneda algebra Ext_{A}(𝕂, 𝕂) is generated in degree 1. One has then Ext_{A}(𝕂, 𝕂) ≅ A^{!}.

Indeed the Yoneda algebra Ext_{S(V)}(𝕂, 𝕂) of the symmetric algebra S(V) is just the exterior algebra

Ext_{S(V)}^{n}(𝕂, 𝕂) = (Tor_{n}^{S(V)}(𝕂, 𝕂))^{∗} = Λ^{n}V^{∗}

which is obviously generated by V^{∗}, i.e., in degree 1, by the wedge product. Through the homotopy transfer the Yoneda algebra Ext_{S(V)}(𝕂, 𝕂) inherits a C_{∞} - structure but it is easy to show (by a degree preserving argument) that the latter C_{∞} -algebra is formal, i.e., all higher multiplications are trivial, m_{n} = 0 for n ≠ 2.

5. Homology of the free 2-nilpotent algebra 𝔤 = V ⊕ Λ^{2}V

Let 𝔤 be the free 2-step nilpotent Lie algebra generated by a vector space V in degree 1, 𝔤 = V ⊕ [V, V]. In other words the Lie bracket of the graded Lie algebra 𝔤 = V ⊕ Λ^{2}V is given by

[u, v] = \{ \begin{matrix}u ∧ v & u, v ∈ V \\ 0 & otherwise \end{matrix} .

We denote the Universal Enveloping Algebra (UEA) U𝔤 by P S and refer to it as parastatistics algebra.^{2} Throughout this note we will consider the generators space V to be an ordinary vector space V which corresponds to a parafermionic algebra P S(V) = U𝔤. The case of a ℤ_{2}-space of generators V = V_{0} ⊕ V_{1}, that is, P S(V) is the Universal Enveloping Algebra of a Lie super-algebra 𝔤 = 𝔤¯0 ⊕ 𝔤¯1 (which would include the parabosonic algebras) will be treated elsewhere. More on parastatistics algebras and their application to combinatorics could be found in the articles [5, 13].

The parastatistics algebra P S(V) generated by a finite dimensional vector space V is the positively graded algebra with degree induced by the tensor degree

P S(V) := U𝔤 = U( V ⊕ ⋀^{2} V ) = T (V)/([[V, V], V]).

We shall write simply P S when the space of generators V is clear from the context.

2Such cubic algebras arise through the exchange relations between the operators in a quantization procedure introduced by Green [8] for particles obeying more general statistics than Bose–Einstein or Fermi–Dirac, coined parabosons and parafermions.

The homologies H_{n}(𝔤, 𝕂) of the free 2-nilpotent Lie algebra 𝔤 are the homologies of the chain complex

⋀^{n} 𝔤 = ⋀^{n} ( V ⊕ ⋀^{2} V ) = ⨁_{s+r=n} ⋀^{s} ( ⋀^{2} V ) ⊗ ⋀^{r} (V)

with differentials ∂_{n} : ⋀ ^{s} ( ⋀ ^{2} V ) ⊗ ⋀ ^{r}(V) → ⋀ ^{s+1} ( ⋀ ^{2} V ) ⊗ ⋀ ^{r−2} (V) given by

∂_{n} : e_{i_{1} j_{1}} ∧ ⋯∧ e_{i_{s} j_{s}} ⊗ e_{l_{1}} ∧ ⋯∧ e_{l_{r}} ↦

∑_{i<j} (−1)^{i+j} e_{l_{i} l_{j}} ∧ e_{i_{1} j_{1}} ∧ ⋯∧ e_{i_{s} j_{s}} ⊗ e_{l_{1}} ∧ ⋯∧ ê_{l_{i}} ∧ ⋯∧ ê_{l_{j}} ∧ ⋯∧ e_{l_{r}} .

The differential ∂ identifies a pair of degree 1 generators e_{i}, e_{j} ∈ V with one degree 2 generator e_{ij} := (e_{i} ∧ e_{j}) = [e_{i}, e_{j}] ∈ Λ^{2}V.

The cohomologies H^{n}(𝔤, 𝕂) arise from the dualized complex with coboundary map δ^{n} : ⋀ ^{n} 𝔤^{∗} → ⋀ ^{n+1} 𝔤^{∗} which is transposed to the differential ∂_{n+1}

δ^{n} : e_{i_{1} j_{1}}^{∗} ∧ ⋯∧ e_{i_{s} j_{s}}^{∗} ⊗ e_{l_{1}}^{∗} ∧ ⋯∧ e_{l_{r}}^{∗} ↦

∑_{k=1}^{s} ∑_{i_{k} <j_{k}} (−1)^{i+j} e_{i_{1} j_{1}}^{∗} ∧ ⋯∧ ê_{i_{k} j_{k}}^{∗} ∧ ⋯∧ e_{i_{s} j_{s}}^{∗} ⊗ e_{i_{k}}^{∗} ∧ e_{j_{k}}^{∗} ∧ e_{l_{1}}^{∗} ∧ ⋯∧ ⋯∧ e_{l_{r}}^{∗} .

In the presence of a metric g one has identifications V ≅^{g} V^{∗} and ⋀ ^{•} 𝔤 ≅^{g} ⋀ ^{•} 𝔤^{∗}. The adjoint operator ∂_{n}^{∗} : ⋀ ^{n} 𝔤 → ⋀ ^{n+1} 𝔤 is defined by g(∂_{n}^{∗}v, w) = g(v, ∂_{n+1} w). One can show that independently of the metric g chosen the action of ∂_{n}^{∗} takes the form

∂_{n}^{∗} : e_{i_{1} j_{1}} ∧ ⋯∧ e_{i_{s} j_{s}} ⊗ e_{l_{1}} ∧ ⋯∧ e_{l_{r}} ↦

∑_{k=1}^{s} ∑_{i_{k} <j_{k}} (−1)^{i+j} e_{i_{1} j_{1}} ∧ ⋯∧ ê_{i_{k} j_{k}} ∧ ⋯∧ e_{i_{s} j_{s}} ⊗ e_{i_{k}} ∧ e_{j_{k}} ∧ e_{l_{1}} ∧ ⋯∧ ⋯∧ e_{l_{r}} .

We will see in the following that after the identification ⋀ ^{•} 𝔤 ≅^{g} ⋀ ^{•} 𝔤^{∗} the map ∂^{∗} =^{g} δ will play the role of homotopy for the chain complex (⋀ ^{•} 𝔤, ∂_{•}), and vice versa: the boundary map ∂ =^{g} δ^{∗} is a homotopy for the cochain complex (⋀ ^{•} 𝔤^{∗}, δ^{•}).

The complexes (⋀ ^{n} 𝔤, ∂_{n}) and (⋀ ^{n} 𝔤^{∗}, δ^{n}) are bigraded by two different degrees; the homological degree n := r + s counting the number of Lie algebra generators and the tensor degree t := 2s+r also called weight. The cohomologies H^{n}(𝔤, 𝕂) can have components of different weight t, H^{n}(𝔤, 𝕂) = ⨁ _{t} H^{n}(𝔤, 𝕂)_{t} and the weight t is in fact the Adams grading on the Yoneda algebra Ext_{U𝔤}^{n}(𝕂, 𝕂)_{t} [15]. The differential and the homotopy, δ = ∂^{∗} and ∂ = δ^{∗} do not alter the weight t, but raise and lower the homological degree n.

The operations m_{k} in the homotopy algebra are bigraded by homological and Adams gradings of bidegree (k, t) = (2−k, 0). The bi-grading imposes the vanishing of many higher products.

5.1. Homology of 𝔤 as a GL(V)-module. A Schur module V_{λ} is an irreducible polynomial GL(V)-module labelled by a Young diagram λ. The basis of a Schur module V_{λ} is in bijection with semistandard Young tableaux with entries in the set {1, …, dimV }. The action of the linear group GL(V) on the space V of the generators of the Lie algebra 𝔤 induces a GL(V)-action on the universal enveloping algebra P S = U𝔤 ≅ S(V ⊕ Λ^{2}V) and on the space ⋀ ^{•} 𝔤 ≅ ⋀ ^{•}(V ⊕ ⋀ ^{2} V).

The maps ∂ and ∂^{∗} both commute with the GL(V)-action. It follows that the homology and cohomology carry structure of GL(V)-modules and hence can be decomposed into irreducibles.

The Laplacian Δ = ⊕_{n⩾0} Δ_{n} is defined to be the self-adjoint operator

Δ_{n} = ∂_{n+1} ∂_{n+1}^{∗} + ∂_{n}^{∗}∂_{n} ∈ End( ⋀^{n} 𝔤) .

Its kernel is a complete set of representatives for the homology classes in H_{n}(𝔤, 𝕂)

kerΔ_{n} ≅ H_{n}(𝔤, 𝕂).

The decomposition of the GL(V)-module H_{n}(𝔤, 𝕂) into irreducible polynomial representations V_{λ} is given by the following theorem.

Theorem 5.1 (Józefiak and Weyman [9], Sigg [16]). The homology H_{•}(𝔤, 𝕂) of the free 2-nilpotent Lie algebra 𝔤 = V ⊕ ⋀ ^{2} V decomposes into a sum of irreducible GL(V)-modules

H_{n}(𝔤, 𝕂) ≅ Tor_{n}^{P S} (𝕂, 𝕂)(V) ≅ ⨁_{λ:λ=λ′} V_{λ} such that n = \frac{1}{2}(|λ| + r(λ)),

where the sum is over the self-conjugate Young diagrams λ, |λ| stands for the number of boxes in λ and r(λ) for the rank of λ (the number of diagonal boxes in λ).

Remark 5.1. The free 2-step nilpotent Lie algebra 𝔤 is the nilradical of a parabolic subalgebra of a simple Lie algebra of type C and its cohomology can be described by a general result of Bertram Kostant [11, Theorem 5.14]. A derivation of the cohomology H^{•}(𝔤, 𝕂) in these lines has been worked out by Grassberger, King and Tirao [7] thus providing one more proof of Theorem 5.1 via the isomorphism H_{n}(𝔤, 𝕂) ≅ Tor_{n}^{P S} (𝕂, 𝕂)(V) ≅ Ext_{P S}^{n} (𝕂, 𝕂)^{∗} ≅ H^{n}(𝔤, 𝕂)^{∗}.

5.2. Homological interpretation of the Littlewood formula. We recall the beautiful result of Józefiak and Weyman [9] giving a representation-theoretic interpretation of the Littlewood formula

∏_{i} (1 − x_{i}) ∏_{i<j} (1 − x_{i}x_{j}) = ∑_{λ:λ=λ′} (−1)^{\frac{1}{2} (|λ|+r(λ))} s_{λ}(x).

Here the sum is over all self-conjugate Young diagrams λ and s_{λ}(x) stands for the Schur function with diagram λ.

One knows that for the graded algebra P S there exists a minimal resolution^{3} by projective modules in the graded category

(5.1)  P_{•} : 0 → P_{d} → ⋯→ P_{n} → ⋯→ P_{2} → P_{1} → P_{0} →^{ϵ} 𝕂 → 0.

Here the length d of the resolution is the projective dimension of the algebra P S which is d = \frac{1}{2} dimV (dimV +1). Since P S is positively graded and, in the category of positively graded modules over connected locally finite graded algebras, projective module is the same as free module [4], we have P_{n} ≅ P S ⊗ E_{n}, where E_{n} are finite dimensional vector spaces. Thus we deal with a minimal resolution of 𝕂 by free P S-modules and the minimality implies that the derived complex 𝕂⊗_{P S} P_{•} has vanishing differentials, i.e., Tor_{•}^{P S} (𝕂, 𝕂) = H_{•}(𝕂 ⊗_{P S} P_{•}) = 𝕂 ⊗_{P S} P_{•}. Then the multiplicity spaces E_{n} = Tor_{n}^{P S} (𝕂, 𝕂) are fixed by Theorem 5.1 and thus the data H_{n}(𝔤, 𝕂) = Tor_{n}^{P S} (𝕂, 𝕂) encodes the minimal free resolution P_{•} (cf. 5.1) which is unique (up to isomorphism).

The Euler characteristics of P_{•} implies an identity about the GL(V)-characters

chP S(V).ch( ⨁_{λ:λ=λ′} (−1)^{\frac{1}{2} (|λ|+r(λ))} V_{λ}) = 1.

The character of a Schur module V_{λ} is the Schur function, chV_{λ} = s_{λ}(x). Due to the Poincaré–Birkhoff–Witt theorem P S(V) ≅ S(V ⊕⋀ ^{2} V) thus the identity reads

∏_{i} \frac{1}{(1 − x_{i})} ∏_{i<j} \frac{1}{(1 − x_{i}x_{j})} ∑_{λ:λ=λ′} (−1)^{\frac{1}{2} (|λ|+r(λ))} s_{λ}(x) = 1.

But the latter identity is nothing but a rewriting of the Littlewood identity (5.1). The moral is that the Littlewood identity reflects a homological property of the algebra P S, namely the above particular structure of the minimal projective (free) resolution of 𝕂 by P S-modules.

5.3. Ext_{P S}^{•} (𝕂, 𝕂) as a C_{∞} -algebra.

Theorem 5.2. The cohomology H^{•}(𝔤, 𝕂) ≅ Ext_{P S}^{•} (𝕂, 𝕂) of the free 2-nilpotent Lie algebra 𝔤 = V ⊕ ⋀ ^{2} V is a homotopy commutative algebra which is generated in degree 1 (i.e., in H^{1}(𝔤, 𝕂)) by the operations m_{2} and m_{3}.

Proof. We start by choosing a metric g on the vector space V and an or-

thonormal basis g(e_{i}, e_{j}) = δ_{ij} . The choice induces a metric on ⋀ ^{•} 𝔤 ≅^{g} ⋀ ^{•} 𝔤^{∗}.

The isomorphisms V ≅ V^{∗} and Tor_{n}^{P S} (𝕂, 𝕂) ≅ Ext_{P S}^{n} (𝕂, 𝕂) and the Theorem 5.1 imply the decomposition of H^{•}(𝔤, 𝕂) into irreducible GL(V)-modules

H^{n}(𝔤, 𝕂) ≅ H^{n}(⋀ 𝔤^{∗}, δ) ≅ Ext_{P S}^{n} (𝕂, 𝕂) ≅ ⨁_{λ:λ=λ′} V_{λ},

where the sum is over all self-conjugate diagrams λ such that n = \frac{1}{2}(|λ| + r(λ)).

3The Chevalley–Eilenberg complex does not provide a minimal resolution of the module 𝕂, in general.

The adjoint of the boundary map ∂, δ :=^{g} ∂^{∗} is the differential in the DGA ( ⋀ 𝔤^{∗}, δ) while δ^{∗} :=^{g} ∂ plays the role of a homotopy. In view of Lemma 2.1 we have the cohomology H^{•}(⋀ ^{•} 𝔤^{∗}, δ^{•}) as deformation retract of the complex ( ⋀ ^{•} 𝔤^{∗}, δ^{•}),

pi = Id_{H^{•} (⋀ ^{•} 𝔤^{∗})} , ip − Id_{⋀ ^{•} 𝔤^{∗}} = δδ^{∗} + δ^{∗}δ, δ^{∗} =^{g} ∂.

Here the projection p identifies the subspace kerδ ∩ kerδ^{∗} with H^{•}(⋀ ^{•} 𝔤^{∗}), which is the orthogonal complement of the space of the coboundaries imδ. The cocyclechoosing homomorphism i is Id on H^{•}(⋀ ^{•} 𝔤^{∗}) and zero on coboundaries.

We apply the Kadeishvili homotopy transfer theorem 2.1 for the commutative DGA (⋀ ^{•} 𝔤^{∗}, μ, δ^{•}) and its deformation retract H^{•}(⋀ ^{•} 𝔤^{∗}) ≅ H^{•}(𝔤, 𝕂) and conclude that the cohomology H^{•}(𝔤, 𝕂) is a C_{∞} -algebra.

The Kontsevich and Soibelman tree representations of the operations m_{n} provide explicit expressions. Let us take μ to be the super-commutative product ∧ on the DGA (⋀ ^{•} 𝔤^{∗}, δ^{•}). The projection p maps onto the Schur modules V_{λ} with self-conjugated Young diagram λ = λ′.

The binary operation on the generators e_{i} ∈ H^{1}(𝔤, 𝕂) is trivial, one gets

m_{2}(e_{i}, e_{j}) = p(e_{i} ∧ e_{j}) = 0 p(V_{(1^{2} )}) = 0.

Hence H^{•}(𝔤, 𝕂) could not be generated in H^{1}(𝔤, 𝕂) as an algebra with the binary product m_{2}.

The ternary operation m_{3} restricted to H^{1}(𝔤, 𝕂) is nontrivial, indeed taking into account the Koszul sign rule we get the following representative cocycles

m_{3}(e_{i}, e_{j} , e_{k}) = p {−e_{i} ∧ ∂(e_{j} ∧ e_{k}) − ∂(e_{i} ∧ e_{j}) ∧ e_{k}}
= p {e_{ij} ∧ e_{k} + e_{i} ∧ e_{jk} } = e_{ij} ∧ e_{k} − e_{jk} ∧ e_{i} ∈ H^{2}(𝔤, 𝕂).

The complete antisymmetrization of the monomial e_{ik} ∧ e_{j} spans the Schur module V_{(1^{3} )} and thus it is projected out, p(e_{ij} ∧ e_{k} + e_{jk} ∧ e_{i} + e_{ki} ∧ e_{j}) = 0. Therefore the monomials e_{ij} ∧ e_{k} modulo V_{(1^{3} )} span a Schur module V_{(2,1)} ≅ H^{2}(𝔤, 𝕂) having the representative cocycles in bijection with the semistandard Young tableaux with diagram (2, 1),

\begin{matrix}e_{ij} ∧ e_{k} − e_{jk} ∧ e_{i} & ↔ & \begin{matrix}i & k \\ j \end{matrix} for i < j, i ⩽ k, \\ e_{jk} ∧ e_{i} − e_{ki} ∧ e_{j} & ↔ & \begin{matrix}i & j \\ k \end{matrix} for pi < k, i ⩽ j. \end{matrix}

We check the symmetry condition on the ternary operation m_{3} in the C_{∞} - algebra; indeed m_{3} vanishes on the (signed) shuffles Sh_{1,2}

m_{3}(e_{i} � e_{j} ⊗ e_{k}) = m_{3}(e_{i}, e_{j} , e_{k}) − m_{3}(e_{j} , e_{i}, e_{k}) + m_{3}(e_{j} , e_{k}, e_{i}) = 0.

Similarly one gets m_{3}(e_{i} ⊗ e_{j} � e_{k}) = 0 on shuffles Sh_{2,1}.

On the level of Schur modules the ternary operation glues three fundamental GL(V)-modules V_{□} into a Schur module V_{(2,1)} . By iteration of the process of gluing boxes we generate all elementary hooks V_{k} := V_{(k+1,1^{k})} ,

m_{3}(V_{□}, V_{□}, V_{□})= V_{{□ \atop □} ; □}

m_{3} ( V_{□}, V_{{□ \atop □} ; □} , V_{□})= V_{□ ; □ ; □ ; □ ; □}
⋯

m_{3}(V_{0}, V_{k}, V_{0})= V_{k+1} .

In our context the more convenient notation for Young diagrams is due to Frobenius: λ := (a_{1}, …, a_{r}|b_{1}, …b_{r}) stands for a diagram λ with a_{i} boxes in the i-th row on the right of the diagonal, and with b_{i} boxes in the i-th column below the diagonal and the rank r = r(λ) is the number of boxes on the diagonal.

For self-dual diagrams λ = λ′, i.e., a_{i} = b_{i} we set V_{a_{1} ,...,a_{r}} := V_{(a_{1} ,...,a_{r} |a_{1} ,...a_{r} )} when a_{1} > a_{2} > ⋯> a_{r} ⩾ 0 (and set the convention V_{a_{1} ,...,a_{r}} := 0 otherwise). Any two elementary hooks V_{a_{1}} and V_{a_{2}} can be glued together by the binary operation m_{2}, the decomposition of m_{2}(V_{a_{1}} , V_{a_{2}} ) ≅ m_{2}(V_{a_{2}} , V_{a_{1}} ) is given by

m_{2}(V_{a_{1}} , V_{a_{2}} ) = V_{a_{1} ,a_{2}} ⊕ ( ⨁_{i=1}^{a_{2}} V_{a_{1} +i,a_{2} −i} ) , a_{1} ⩾ a_{2}

where the “leading” term V_{a_{1} ,a_{2}} has the diagram with minimal height. Hence any m_{2}-bracketing of the hooks V_{a_{1}} , V_{a_{2}} , …, V_{a_{r}} yields^{4} a sum of GL(V)-modules

m_{2}(…m_{2}(m_{2}(V_{a_{1}} , V_{a_{2}} ), V_{a_{3}} ), …, V_{a_{r}} ) = V_{a_{1} ,...,a_{r}} ⊕ ⋯

whose module with minimal height is precisely V_{a_{1} ,...,a_{r}} . We conclude that all elements in the C_{∞} -algebra H^{•}(𝔤, 𝕂) can be generated in H^{1}(𝔤, 𝕂) by m_{2} and m_{3}.

□

One could draw a parallel between the theorem for the cubic algebra P S and the Proposition 4.1 for the Koszul algebra; in both cases the Yoneda algebra Ext_{P S}^{•} (𝕂, 𝕂) is generated only in Ext_{P S}^{1} (𝕂, 𝕂). Although we have the notion of N-Koszul algebras for the N-homogeneous algebras [2, 3], it turns out that the cubic algebra P S is not 3-Koszul, beside the exceptional case when dimV = 2. Instead the algebra P S = U𝔤 falls in the class of Artin–Schelter-regular algebras [1], being an UEA of positively graded Lie algebra (for a proof see [6]). The parallel between the quadratic Koszul algebra S(V) and the cubic AS-regular regular algebra P S(V) suggests that the C_{∞} -algebra Ext_{P S}^{•} (𝕂, 𝕂) is a generalization of a Koszul dual algebra of P S in the realm of the homotopy algebras, an idea that has been put forward in [15].

The analogy would be complete if we had the following conjectural proposition.

Proposition 5.1. The cohomology H^{•}(𝔤, 𝕂) ≅ Ext_{P S}^{•} (𝕂, 𝕂) of the free 2nilpotent Lie algebra 𝔤 = V ⊕ ⋀ ^{2} V can be endowed with a structure of C_{∞} -algebra having trivial higher multiplications m_{k} = 0, k ⩾ 4.

4The operation m_{2} is associative thus the result does not depend on the choice of the bracketing.

So far we have been able to prove this conjecture only in dimensions dimV ⩽ 3. Our proof rests entirely on the bigrading (2 − k, 0) of the multiplication m_{k} by homological and tensor degree in the C_{∞} -algebra Ext_{P S}^{•} (𝕂, 𝕂). The bigrading arguments work only for dimV = 2 and dimV = 3 thus for a complete proof the conjecture would need more refined methods.

Acknowledgements. We are grateful to Jean-Louis Loday for many enlightening discussions and his encouraging interest. Todor Popov thanks the Serbian hosts for the warm hospitality, the financial support and for the stimulating atmosphere during the conference in Zlatibor.

References

1. M. Artin, W. F. Schelter, Graded algebras of global dimension 3, Adv. Math. 66 (1987), 171–216.

2. R. Berger, Koszulity for nonquadratic algebras, J. Algebra 239 (2001), 705–734.

3. R. Berger, M. Dubois-Violette, M. Wambst, Homogeneous algebras, J. Algebra 261 (2003), 172–185.

4. H. Cartan, Homologie et cohomologie d’ une algèbre graduée, Séminaire Henri Cartan 11 (1958), 1–20.

5. M. Dubois-Violette, T. Popov, Homogeneous Algebras, Statistics and Combinatorics, LMP 61 (2002), 159–170.

6. G. Floystad, J. E. Vayne, Artin-Schelter regular algebras of dimension five, Algebra, Geometry, and Mathematical Physics, Banach Center Publications, 93 (2011), 19–39.

7. J. Grassberger, A. King, P. Tirao, On the homology of free 2-step nilpotent Lie algebras, J. Algebra 254 (2002), 213–225.

8. H. S. Green, A Generalized method of field quantization, Phys. Rev. 90 (1953), 270–273.

9. T. Józefiak, J. Weyman, Representation-theoretic interpretation of a formula of D. E. Littlewood, Math. Proc. Cambridge Phil. Soc. 103 (1988), 193–196.

10. T. Kadeishvili, The A_{∞} -algebra structure and cohomology of Hochschild and Harrison, Proc. Tbil. Math. Inst. 91 (1988), 19–27.

11. B. Kostant, Lie algebra cohomology and the generalized Borel-Weil theorem, Annals of Math. 74 (1961), 329–387.

12. M. Kontsevich, Y. Soibelman, Deformations of algebras over operads and the Deligne conjecture, Math. Phys. Stud. 21 (2000), Kluwer, Dordrecht, 255–307.

13. J.-L. Loday, T. Popov, Parastatistics algebra, Young tableaux and the super plactic monoid, Int. J. Geom. Meth. Mod. Phys. 5 (2008), 1295–1314.

14. J.-L. Loday, B. Vallette, Algebraic Operads, Grundlehren Math. Wiss. 346, Springer, Heidelberg, 2012.

15. D.-M. Lu, J. H. Palmieri, Q.-S. Wu, J. J. Zhang, A_{∞} -algebras for ring theorists, Algebra Colloquium 11 (2004), 91–128.

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

Laboratoire de Physique Theorique, UMR 8627 Universite Paris XI, Batiment 210, F-91 405 Orsay Cedex, France

 Michel.Dubois-Violette@th.u-psud.fr

Theoretical Physics Division, Institute for Nuclear Research and Nuclear Energy Bulgarian Academy of Sciences, Sofia, Bulgaria

 tpopov@inrne.bas.bg