Cyclic Cohomology of Lie Algebras

Bahram Rangipour and Serkan Sütlü

Received: September 13, 2011 Revised: May 11, 2012

Communicated by Max Karoubi

Abstract. We define and completely determine the category of Yetter-Drinfeld modules over Lie algebras. We prove a one to one correspondence between Yetter-Drinfeld modules over a Lie algebra and those over the universal enveloping algebra of the Lie algebra. We associate a mixed complex to a Lie algebra and a stable-Yetter-Drinfeld module over it. We show that the (truncated) Weil algebra, the Weil algebra with generalized coefficients defined by Alekseev-Meinrenken, and the perturbed Koszul complex introduced by Kumar-Vergne are examples of such a mixed complex.

2010 Mathematics Subject Classification: 17B56, 16T15, 16E45, 19D55 Keywords and Phrases:

Yetter-Drinfeld modules, Lie algebras, mixed complex, Weil algebra, Koszul complex, Hopf cyclic cohomology

Contents

1 Introduction  484
2 The model complex for G-differential algebras  485
2.1 G-differential algebras . . . . . . . . . . . . . . . . . . . . . . . 485
2.2 The model complex . . . . . . . . . . . . . . . . . . . . . . . . . 487
3 Lie algebra cohomology and Perturbed Koszul complex 488
3.1 Lie algebra cohomology . . . . . . . . . . . . . . . . . . . . . . 489
3.2 Perturbed Koszul complex . . . . . . . . . . . . . . . . . . . . . 489
3.3 Weyl algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . 491

4 Lie algebra homology and Poincaré duality  492
4.1 Lie algebra homology . . . . . . . . . . . . . . . . . . . . . . . . 492
4.2 Poincaré duality . . . . . . . . . . . . . . . . . . . . . . . . . . 493
5 Lie algebra coaction and SAYD coefficients  496
5.1 SAYD modules and cyclic cohomology of Hopf algebras …. 496
5.2 SAYD modules over Lie algebras . . . . . . . . . . . . . . . . . 498
5.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 507
6 Cyclic cohomology of Lie algebras  512

1 Introduction

One of the well-known complexes in mathematics is the Chevalley-Eilenberg complex of a Lie algebra 𝔤 with coefficients in a 𝔤-module V [2].

\xymatrix{C^{•}(𝔤, V) : & V \ar[r]^{d_{CE}} & V ⊗ 𝔤^{∗} \ar[r]^{d_{CE}} & V ⊗ ∧^{2}𝔤^{∗} \ar[r]^{d_{CE}} & ⋯}  (1.1)

Through examples, we can see that when the coefficients space V is equipped with more structures, then the complex (C^{•}(𝔤, V), d_{CE}), together with another operator d_{K} : C^{•}(𝔤, V) → C^{•−1}(𝔤, V), called Koszul boundary, turns into a mixed complex. That is d_{CE} + d_{K} defines a coboundary on the total complex

W^{•} = ⨁_{•≥p≥0} C^{2p−•}(𝔤, V).

Among examples, one observes that,

• the well-known (truncated) Weil complex is achieved by V := S(𝔤^{∗})_{[2q]} the (truncated) polynomial algebra of 𝔤,

• the Weil algebra with generalized coefficients defined by Alekseev-Meinrenken in [1] is obtained by V := E′(𝔤^{∗}), the convolution algebra of compactly supported distributions on 𝔤^{∗},

• finally it was shown by Kumar-Vergne that if V is a module over the Weyl algebra D(𝔤) then (W^{•}, d_{CE} + d_{K}) is a complex which is called perturbed Koszul complex [14].

In this paper we prove that (W^{•}, d_{CE} + d_{K}) is a complex if and only if V is a unimodular stable module over the Lie algebra 𝔤̃, where 𝔤̃ := 𝔤^{∗} >◁ 𝔤 is the semidirect product Lie algebra 𝔤^{∗} and 𝔤. Here 𝔤^{∗} := Hom(𝔤, ℂ) is thought of as an abelian Lie algebra acted upon by the Lie algebra 𝔤 via the coadjoint representation.

Next, we show that any Yetter-Drinfeld module over the enveloping Hopf algebra U(𝔤) yields a module over 𝔤̃ and conversely any locally conilpotent module over 𝔤̃ amounts to a Yetter-Drinfeld module over the Hopf algebra U(𝔤). This correspondence is accompanied with a quasi-isomorphism which reduces to the antisymmetrization map if the module V is merely a 𝔤-module. The isomorphism generalizes the computation of the Hopf cyclic cohomology of U(𝔤) in terms of the Lie algebra homology of 𝔤 carried out by Connes-Moscovici in [4].

Throughout the paper, 𝔤 denotes a finite dimensional Lie algebra over ℂ, the field of complex numbers. We denote by X_{1}, …, X_{N} and θ^{1}, …, θ^{N} a dual basis for 𝔤 and 𝔤^{∗} respectively, and by δ ∈ 𝔤^{∗} the trace of the adjoint representation of 𝔤 on itself. All tensor products are over ℂ.

B. R. would like to thank Alexander Gorokhovsky for the useful discussions on the G-differential algebras, and is also grateful to the organizers of NCGOA 2011 at Vanderbilt University, where these discussions took place.

2 The model complex for G-differential algebras

In this section we first recall G-differential algebras and their basic properties. Then we introduce our model complex which is the main motivation of this paper. The model complex includes as examples Weil algebra and their truncations, perturbed Koszul complex introduced by Kumar- Vergne in [14], and Weil algebra with generalized coefficients introduced by Alekseev-Meinrenken [1].

2.1 G-differential algebras

Let ĝ = 𝔤_{−1} ⊕ 𝔤_{0} ⊕ 𝔤_{1} be a graded Lie algebra, where 𝔤_{−1} and 𝔤_{0} are Ndimensional vector spaces with bases ι_{1}, ⋯, ι_{N}, and L_{1}, ⋯, L_{N} respectively, and 𝔤_{1} is generated by d.

We let C_{jk}^{i} denote the structure constants of the Lie algebra 𝔤_{0} and assume that the graded-bracket on ĝ is defined as follows.

[ι_{p}, ι_{q}] = 0,  (2.1)
[L_{p}, ι_{q}] = C_{pq}^{r}ι_{r},  (2.2)
[L_{p}, L_{q}] = C_{pq}^{r}L_{r},  (2.3)
[d, ι_{k}] = L_{k},  (2.4)
[d, L_{k}] = 0,  (2.5)
[d, d] = 0.  (2.6)

Now let G be a (connected) Lie group with Lie algebra 𝔤. We assume 𝔤̂ be as above with 𝔤_{0} ≅ 𝔤 as Lie algebras.

A graded algebra A is called a G-differential algebra if there exists a representation ρ : G → Aut(A) of the group G and a graded Lie algebra homomorphism ρ̂: ĝ → End(A) compatible in the following way:

\frac{d}{dt}|_{t=0} ρ(exp(tX)) = ρ̂(X)  (2.7)
ρ(a)ρ̂(X)ρ(a^{−1}) = ρ̂(Ad_{a}X)  (2.8)
ρ(a)ι_{X}ρ(a^{−1}) = ι_{Ad_{a}X}  (2.9)
ρ(a)dρ(a^{−1}) = d  (2.10)

for any a ∈ G and any X ∈ 𝔤. For further discussion on G-differential algebras we refer the reader to [8, chapter 2] and [1].

The exterior algebra ⋀ 𝔤^{∗} and the Weil algebra are examples of G-differential algebras.

Here we recall W (𝔤), the Weil algebra of a finite dimensional Lie algebra 𝔤, by

W (𝔤) = ⋀ 𝔤^{∗} ⊗ S(𝔤^{∗}),

with the grading

W (𝔤) = ⨁_{l≥0} W^{l}(𝔤),  (2.11)

where

W^{l}(𝔤) = ⨁_{p+2q=l} W^{p,q},  W^{p,q} := ∧^{p}𝔤^{∗} ⊗ S^{q}(𝔤^{∗}).  (2.12)

It is equipped with two degree +1 differentials as follows. The first one is

\begin{matrix}d_{K} : ∧^{p}𝔤^{∗} ⊗ S^{q}(𝔤^{∗}) → ∧^{p−1}𝔤^{∗} ⊗ S^{q+1}(𝔤^{∗}) \\ φ ⊗ R ↦ ∑_{j} ι_{X_{j}}(φ) ⊗ Rθ^{j} \end{matrix}  (2.13)

and it is called the Koszul coboundary. The second one is the Chevalley-Eilenberg coboundary (Lie algebra cohomology coboundary)

d_{CE} : ∧^{p}𝔤^{∗} ⊗ S^{q}(𝔤^{∗}) → ∧^{p+1}𝔤^{∗} ⊗ S^{q}(𝔤^{∗}).  (2.14)

Then d_{CE} + d_{K} : W^{l}(𝔤) → W^{l+1}(𝔤) equips W (𝔤) with a differential graded algebra structure. It is known that via coadjoint representation W (𝔤) is a G-differential algebra.

A G-differential algebra is called locally free if there exists an element

Θ = ∑_{i} X_{i} ⊗ θ^{i} ∈ (𝔤 ⊗ A^{odd})^{G}

called the algebraic connection form.

We assume that Θ ∈ (𝔤 ⊗ A^{1})^{G}, and we have

ι_{k}(Θ) = X_{k}, and L_{k}(θ^{i}) = −C_{kl}^{i}θ^{l}.

2.2 The model complex

Let (A, Θ) be a locally free G-differential algebra with dim(G) = N. We assume that V is a vector space with elements L_{k} and L^{k} in End(V), 1 ≤ k ≤ N.

We consider the graded space A ⊗ V with the grading induced from that of A. Using all information of the G-differential algebra structure of A and the connection form Θ ∈ (𝔤 ⊗ A^{1})^{G}, we introduce the map

D(x ⊗ v) := d(x) ⊗ v + θ^{k}x ⊗ L_{k}(v) + ι_{k}(x) ⊗ L^{k}(v)  (2.15)

as a sum of a degree +1 map and a degree −1 map.

Proposition 2.1. Let (A, Θ) be a locally free G-differential algebra. Then the map

d_{K}(x ⊗ v) = ι_{k}(x) ⊗ L^{k}(v)  (2.16)

is a differential, that is d_{K}^{2} = 0, if and only if V is a ℂ^{N}-module via L^{k}s, i.e.,

[L^{p}, L^{q}] = 0, 1 ≤ p, q, ≤ N.

Proof. Assume that [L^{j}, L^{i}] = 0. Then

d_{K} ∘ d_{K}(x ⊗ v) = ι_{l}ι_{k}(x) ⊗ L^{l}L^{k}(v) = 0  (2.17)

by the commutativity of L_{k}s and the anti-commutativity of ι_{k}s.

Conversely, if d_{K} has the property d_{K} ∘ d_{K} = 0, then by using ι_{k}(θ^{j}) = δ_{k}^{j} we have

d_{K} ∘ d_{K}(θ^{i}θ^{j} ⊗ v) = d_{K}(θ^{j} ⊗ L^{i}(v) − θ^{i} ⊗ L^{j}(v)) = 1 ⊗ [L^{j}, L^{i}](v) = 0 (2.18)

which implies [L^{j}, L^{i}] = 0.

Definition 2.2. [8]. For a commutative locally free G-differential algebra A, the element Ω = ∑_{i} Ω^{i} ⊗ X_{i} ∈ (A^{2} ⊗ 𝔤)^{G}, satisfying

d(θ^{i}) = −\frac{1}{2}C_{pq}^{i}θ^{p}θ^{q} + Ω^{i},  (2.19)

is called the curvature of the connection Θ = ∑_{i} θ^{i} ⊗ X_{i}.

We call a commutative locally free G-differential algebra (A, Θ) flat if Ω = 0, or equivalently

d(θ^{k}) = −\frac{1}{2}C_{pq}^{k}θ^{p}θ^{q}.  (2.20)

Proposition 2.3. Let (A, Θ) be a commutative locally free flat G-differential algebra. Then the map

d_{CE}(x ⊗ v) = d(x) ⊗ v + θ^{k}x ⊗ L_{k}(v)  (2.21)

is a differential, that is d_{CE}^{2} = 0, if and only if V is a 𝔤-module via L_{k}, that is [L_{t}, L_{l}] = C_{tl}^{k}L_{k}.

Proof. Using the commutativity of A we see that

\begin{aligned}d_{CE} ∘ d_{CE}(1 ⊗ v) & = ∑_{k} d_{CE}(θ^{k} ⊗ L_{k}(v)) \\ & = ∑_{k} d(θ^{k}) ⊗ L_{k}(v) + ∑_{k,t} θ^{t}θ^{k} ⊗ L_{t}L_{k}(v) \\ & = − ∑_{l,t} \frac{1}{2}C_{tl}^{k}θ^{t}θ^{l} ⊗ L_{k}(v) + ∑_{t,l} \frac{1}{2}θ^{t}θ^{l} ⊗ [L_{t}, L_{l}](v), \end{aligned} (2.22)

which proves the claim.

One notes that we use the commutativity of the differential graded algebra in the above proposition. It would be interesting if one finds an argument for a similar proposition to cover the noncommutative and twisted noncommutative Weil algebra as discussed in [1] and [3] respectively.

Considering the dual 𝔤^{∗} of the Lie algebra 𝔤 as a commutative Lie algebra, we can define the Lie bracket on 𝔤̃ := 𝔤^{∗} >◁ 𝔤 by

[α ⊕ X , β ⊕ Y ] := (L_{X}(β) − L_{Y} (α)) ⊕ [X , Y ].  (2.23)

Accordingly, the next proposition determines the necessary and sufficient conditions on V for (A ⊗ V, d_{CE} + d_{K}) to be a differential complex.

Proposition 2.4. Let A be a commutative locally free flat G-differential algebra and V a 𝔤-module via L_{k}s and a ℂ^{N}-module via L^{k}s. Then, (A ⊗ V, D) is a differential complex if and only if V is

unimodular stable  ∑_{k} L^{k}L_{k} = 0,  (2.24)

and

𝔤̃-module  [L^{i}, L_{j}] = ∑_{k} C_{jk}^{i}L^{k}.  (2.25)

Proof. Once we have d_{CE}^{2} = 0 = d_{K}, then (A ⊗ V, d_{CE} + d_{K}) is a differential complex i.e.,

0 = d_{CE} ∘ d_{K} + d_{K} ∘ d_{CE}

= L_{k} ⊗ L^{k} + θ^{k}ι_{t} ⊗ [L_{k}, L^{t}] + Id ⊗L^{k}L_{k}, (2.26)

if and only if (2.24) and (2.25) hold.

3 Lie algebra cohomology and Perturbed Koszul complex

In this section we specialize the model complex (A ⊗ V, D) defined in (2.15) for A = ⋀ 𝔤^{∗}. We show that the perturbed Koszul complex defined in [14] is an example of the model complex. As another example of the model complex, we cover the Weil algebra with generalized coefficients introduced in [1].

3.1 Lie algebra cohomology

Let 𝔤 be a finite dimensional Lie algebra and V be a right 𝔤-module. Let also {θ^{i}} and {X_{i}} be dual bases for 𝔤^{∗} and 𝔤. The Chevalley-Eilenberg complex C(𝔤, M) is defined by

\xymatrix{V \ar[r]^{d_{CE}} & C^{1}(𝔤, V) \ar[r]^{d_{CE}} & C^{2}(𝔤, V) \ar[r]^{d_{CE}} & ⋯,}  (3.1)

where C^{q}(𝔤, V) = Hom(∧^{q}𝔤, V) is the vector space of all alternating linear maps on 𝔤^{⊗q} with values in V. If α ∈ C^{q}(𝔤, V), then

\begin{matrix}d_{CE}(α)(X_{0}, …, X_{q}) = ∑_{i<j}(−1)^{i+j}α([X_{i}, X_{j}], X_{0} … X̂_{i}, …, X̂_{j}, …, X_{q})+ \\ ∑_{i}(−1)^{i+1}α(X_{0}, …, X̂_{i}, … X_{q})X_{i}. \end{matrix} (3.2)

Alternatively, we may identify C^{q}(𝔤, V) with ∧^{q}𝔤^{∗} ⊗ V and the coboundary d_{CE} with the following one

\begin{matrix}d_{CE}(v) = −θ^{i} ⊗ v · X_{i}, \\ d_{CE}(β ⊗ v) = d_{dR}(β) ⊗ v − θ^{i} ∧ β ⊗ v · X_{i}. \end{matrix}  (3.3)

where d_{dR} : ∧^{p}𝔤^{∗} → ∧^{p+1}𝔤^{∗} is the de Rham derivation defined by d_{dR}(θ^{i}) = \frac{−1}{2}C_{jk}^{i}θ^{j}θ^{k}. We denote the cohomology of (C^{•}(𝔤, V), d_{CE}) by H^{•}(𝔤, V) and refer to it as the Lie algebra cohomology of 𝔤 with coefficients in V.

3.2 Perturbed Koszul complex

With the same assumptions on 𝔤 and V in the previous subsection, we specialize the model complex A ⊗ V defined in (2.15) for A = ⋀ 𝔤^{∗}. Indeed we have W^{n}(𝔤, V) := ∧^{n}𝔤^{∗} ⊗ V, for n ≥ 0, with differentials d_{CE} : W^{n}(𝔤, V) → W^{n+1}(𝔤, V) defined in (3.3) and

\begin{matrix}d_{K} : W^{n}(𝔤, V) → W^{n−1}(𝔤, V) \\ α ⊗ v ↦ ∑_{i} ι_{X_{i}}(α) ⊗ v ⊲ θ^{i}. \end{matrix}  (3.4)

An immediate example is the (truncated) Weil algebra.

Example 3.1 (Weil algebra). Let 𝔤 be a (finite dimensional) Lie algebra and set V = S(𝔤^{∗}) - the polynomial algebra on 𝔤. Then V is a unimodular stable right 𝔤-module via the (co)adjoint action of 𝔤 on 𝔤^{∗} and the initial multiplication of the symmetric algebra as the action of 𝔤^{∗}.

Example 3.2 (Truncated Weil algebra). Let V = S(𝔤^{∗})_{[2n]} be the truncated polynomial algebra on 𝔤. With the same structure as it is defined in Example 3.1 one obtains the differential complex W (𝔤, S(𝔤^{∗})_{[2n]}).

To be able to interpret the coefficient space further, we introduce the crossed product algebra

D̃(𝔤) := S(𝔤^{∗}) >◁ U(𝔤).  (3.5)

In the next proposition, by 𝔤̃ we mean 𝔤^{∗} >◁ 𝔤 with the Lie bracket defined in (2.23).

Proposition 3.3. The algebras D̃(𝔤) and U(𝔤^{∗} >◁ 𝔤) are isomorphic.

Proof. It is a simple case of [15, Theorem 7.2.3], that is

U(𝔤^{∗} >◁ 𝔤) = U(𝔤^{∗}) >◁ U(𝔤) = S(𝔤^{∗}) >◁ U(𝔤) = D̃(𝔤)  (3.6)

Next, we recall the compatibility for a module over a crossed product algebra, for a proof see [16, Lemma 3.6].

Lemma 3.4. Let H be a Hopf algebra, and A an H-module algebra. Then V is a right module on the crossed product algebra A >◁ H if and only if V is a right module on A and a right module on H such that

(v · h) · a = (v · (h_{(1)} ⊳ a)) · h_{(2)}  (3.7)

Finally we restate Proposition 2.4 in the case of A = ⋀ 𝔤^{∗}.

Proposition 3.5. The graded space (W^{•}(𝔤, V), d_{CE} + d_{K}) is a complex if and only if V is a unimodular stable right 𝔤̃-module.

Example 3.6 (Weil algebra with generalized coefficients [1]). Let E′(𝔤^{∗}) be the convolution algebra of compactly supported distributions on 𝔤^{∗}. The symmetric algebra S(𝔤^{∗}) is canonically identified with the subalgebra of distributions supported at the origin. This immediately results with a natural S(𝔤^{∗})-module structure on E′(𝔤^{∗}) via its own multiplication.

Regarding the coordinate functions μ_{i}, 1 ≤ i ≤ N as multiplication operators, we also have [μ_{i}, θ^{j}] = δ_{j}^{i}.

The Lie derivative is described as follows.

L_{i} = C_{ij}^{k}θ^{j}μ_{k}, 1 ≤ i ≤ N.  (3.8)

Therefore,

τ : 𝔤 → End(E′(𝔤^{∗})), X_{i} ↦ C_{ji}^{k}μ_{k}θ^{j} = −L_{X_{i}} − δ(X_{i})I  (3.9)

is a map of Lie algebras, and hence equips E′(𝔤^{∗}) with a right 𝔤-module structure.

We first observe that

∑_{i}(v · X_{i}) ⊲ θ^{i} = C_{ji}^{k}vμ_{k}θ^{j}θ^{i} = 0,  (3.10)

by the commutativity of S(𝔤^{∗}) and the anti-commutativity of the lower indices of the structure coefficients.

Secondly we observe

\begin{matrix}(v · X_{i}) ⊲ θ^{t} = C_{ji}^{k}vμ_{k}θ^{j}θ^{t} = \\ C_{ji}^{k}vθ^{t}μ_{k}θ^{j} + C_{ji}^{t}vθ^{j} = (v ⊲ θ^{t}) · X_{i} + v ⊲ (X_{i} ⊳ θ^{t}), \end{matrix}  (3.11)

i.e., E′(𝔤^{∗}) is a right module over S(𝔤^{∗}) >◁ U(𝔤). Hence we have the complex W (𝔤, E′(𝔤^{∗})).

Finally we remark that in [1] the authors consider compact groups and their Lie algebras, which are unimodular and hence δ = 0. So, their and our actions of 𝔤 coincide.

3.3 Weyl algebra

Following [17] Appendix 1, let V be a (finite dimensional) vector space with dual V^{∗}. Let 𝒫(V) be the algebra of all polynomials on V and S(V) the symmetric algebra on V. Let us use the notation D(V) for the algebra of differential operators on V with polynomial coefficients - the Weyl algebra on V. For any v ∈ V we introduce the operator

∂_{v}(f)(w) := \frac{d}{dt}|_{t=0} f(w + tv).  (3.12)

As a result, we get an injective algebra map v ↦ ∂_{v} ∈ D(V). As a differential operator on V, ∂_{v} is identified with the derivative with respect to v^{∗} ∈ V^{∗}.

Using the bijective linear map 𝒫(V) ⊗ S(V) → D(V) defined as f ⊗ v ↦ fv, and the fact that 𝒫(V) ≅ S(V^{∗}), we conclude that D(V) ≅ S(V^{∗}) ⊗ S(V) as vector spaces.

Following [7], the standard representation of D(V) is as follows. Let {v_{1}^{∗}, ⋯, v_{n}^{∗}} be a basis of V^{∗}. Then, forming E = ℂ[v_{1}^{∗}, ⋯, v_{n}^{∗}], we consider the operators P_{i} ∈ End(E) as ∂/∂v_{i}^{∗} and Q^{i} ∈ End(E) as multiplication by v_{i}^{∗}. Then the relations are

\begin{matrix}[P_{i}, Q^{i}] = I,  [P_{i}, Q^{j}] = 0, i ≠ j \\ [P_{i}, P_{j}] = 0,  [Q^{i}, Q^{j}] = 0, ∀ i, j \end{matrix}  (3.13)

It is observed that if V is a module over D(𝔤) then (W^{•}(𝔤, V), d_{CE} + d_{K}) is a differential complex [14]. We now briefly remark the relation of this result with our interpretation of the coefficient space (2.15). To this end, we first notice that if V is a right module over the Weyl algebra D(𝔤), then it is module over the Lie algebra 𝔤 via the Lie algebra map

τ : 𝔤 → D(𝔤), X_{i} ↦ C_{ki}^{l}P_{l}Q^{k}.  (3.14)

Explicitly, we define the action of the Lie algebra as

v · X_{k} = vτ(X_{k}).  (3.15)

On the other hand, V is also a module over the symmetric algebra S(𝔤^{∗}) via

v ⊲ θ^{k} = vQ^{k}.  (3.16)

Lemma 3.7. Let V be a right module over D(𝔤). Then V is unimodular stable.

Proof. We immediately observe that

∑_{i}(v · X_{i}) ⊲ θ^{i} = ∑_{i} v · (τ(X_{i})Q^{i}) = ∑_{i,l,k} v · (C_{li}^{k}P_{k}Q^{l}Q^{i}) = 0  (3.17)

by the commutativity of Qs and the anti-commutativity of the lower indices of the structure coefficients.

Let us now introduce the map

Φ : D̃(𝔤) → D(𝔤), θ^{j} >◁ X_{i} ↦ Q^{j}τ(X_{i}) = C_{ki}^{l}Q^{j}P_{l}Q^{k}.  (3.18)

Lemma 3.8. The map Φ : D̃(𝔤) → D(𝔤) is well-defined.

Proof. It is enough to prove Φ(X_{i})Φ(θ^{j}) = Φ(X_{i}_{(1)} ⊳ θ^{j})Φ(X_{i}_{(2)}). To this, we observe

\begin{matrix}RHS = Φ(X_{i}_{(1)} ⊳ θ^{j})Φ(X_{i}_{(2)}) = −C_{ik}^{j}Q^{k} + C_{ki}^{l}Q^{j}P_{l}Q^{k} \\ = C_{ki}^{l}P_{l}Q^{j}Q^{k} = C_{ki}^{l}P_{l}Q^{k}Q^{j} = Φ(X_{i})Φ(θ^{j}) = LHS. \end{matrix}  (3.19)

Corollary 3.9. If V is a right module over D(𝔤), then V is a right module over D̃(𝔤) = S(𝔤^{∗}) >◁ U(𝔤).

4 Lie algebra homology and Poincaré duality

In this section, for any Lie algebra 𝔤 and any stable 𝔤̃-module V we define a complex dual to the model complex and we establish a Poincaré duality between these two complexes. The need for this new complex will be justified in the next sections.

4.1 Lie algebra homology

Let 𝔤 be a Lie algebra and V be a right 𝔤-module. We recall the Lie algebra homology complex C_{q}(𝔤, V) = ∧^{q}𝔤 ⊗ V by

\xymatrix{⋯ \ar[r]^{∂_{CE}} & C_{2}(𝔤, V) \ar[r]^{∂_{CE}} & C_{1}(𝔤, V) \ar[r]^{∂_{CE}} & V}  (4.1)

where

\begin{matrix}∂_{CE}(X_{0} ∧ ⋯ ∧ X_{q−1} ⊗ v) = ∑_{i}(−1)^{i}X_{0} ∧ ⋯ ∧ X̂_{i} ∧ ⋯ ∧ X_{q−1} ⊗ v · X_{i}+ \\ ∑_{i<j}(−1)^{i+j}[X_{i}, X_{j}] ∧ X_{0} ∧ ⋯ ∧ X̂_{i} ∧ ⋯ ∧ X̂_{j} ∧ ⋯ ∧ X_{q−1} ⊗ v \end{matrix} (4.2)

We call the homology of the complex (C_{•}(𝔤, V), ∂_{CE}) the Lie algebra homology of 𝔤 with coefficients in V and denote it by H_{•}(𝔤, V).

4.2 Poincaré duality

Let V to be a right 𝔤-module and right S(𝔤^{∗})-module. We consider the graded vector space C_{n}(𝔤, V) := ∧^{n}𝔤 ⊗ V with two differentials ∂_{CE} : C_{n+1}(𝔤, V) → C_{n}(𝔤, V) defined in (4.2), and

∂_{K} : C_{n}(𝔤, V) → C_{n+1}(𝔤, V), Y_{1} ∧⋯∧Y_{n} ⊗v ↦ ∑_{i} X_{i} ∧Y_{1} ∧⋯∧Y_{n} ⊗v ⊲θ^{i}

(4.3)

Let us first justify that ∂_{K} is a differential.

Lemma 4.1. We have ∂_{K} ∘ ∂_{K} = 0.

Proof. We observe that by the commutativity of S(𝔤^{∗}) and the anticommutativity of the wedge product we have

\begin{matrix}∂_{K} ∘ ∂_{K}(Y_{1} ∧ ⋯ ∧ Y_{n} ⊗ v) = ∑_{i} ∂_{K}(X_{i} ∧ Y_{1} ∧ ⋯ ∧ Y_{n} ⊗ v ⊲ θ^{i}) \\ = ∑_{i,j} X_{j} ∧ X_{i} ∧ Y_{1} ∧ ⋯ ∧ Y_{n} ⊗ v ⊲ θ^{i}θ^{j} = 0. \end{matrix}  (4.4)

We say that a right 𝔤̃-module V is stable if

∑_{i}(v ⊲ θ^{i}) · X_{i} = 0.  (4.5)

Proposition 4.2. The space (C_{•}(𝔤, V), ∂_{CE} + ∂_{K}) is a differential complex if and only if V is stable right 𝔤̃-module.

Proof. On the one hand we have

\begin{matrix}∂_{CE}(∂_{K}(Y_{0} ∧ ⋯ ∧ Y_{n} ⊗ v)) \\ = ∑_{i} ∂_{CE}(X_{i} ∧ Y_{0} ∧ ⋯ ∧ Y_{n} ⊗ v ⊲ θ^{i}) \\ = ∑_{i} Y_{0} ∧ ⋯ ∧ Y_{n} ⊗ (v ⊲ θ^{i}) · X_{i} \\ + ∑_{i,j}(−1)^{j+1}X_{i} ∧ Y_{0} ∧ ⋯ ∧ Ŷ_{j} ∧ ⋯ ∧ Y_{n} ⊗ (v ⊲ θ^{i}) · Y_{j} \\ + ∑_{i,j}(−1)^{j+1}[X_{i}, Y_{j}] ∧ Y_{0} ∧ ⋯ ∧ Ŷ_{j} ∧ ⋯ ∧ Y_{n} ⊗ v ⊲ θ^{i} \\ + ∑_{i,j}(−1)^{j+k}[Y_{j}, Y_{k}] ∧ X_{i} ∧ Y_{0} ∧ ⋯ ∧ Ŷ_{j} ∧ ⋯ ∧ Ŷ_{k} ∧ ⋯ ∧ Y_{n} ⊗ v ⊲ θ^{i}, \end{matrix} (4.6)

and on the other hand

\begin{matrix}∂_{K}(∂_{CE}(Y_{0} ∧ ⋯ ∧ Y_{n} ⊗ v)) \\ = ∑_{j}(−1)^{j}∂_{K}(Y_{0} ∧ ⋯ ∧ Ŷ_{j} ∧ ⋯ ∧ Y_{n} ⊗ v · Y_{j}) \\ + ∑_{j,k}(−1)^{j+k}∂_{K}([Y_{j}, Y_{k}] ∧ Y_{0} ∧ ⋯ ∧ Ŷ_{j} ∧ ⋯ ∧ Ŷ_{k} ∧ ⋯ ∧ Y_{n} ⊗ v) \\ = ∑_{i,j}(−1)^{j}X_{i} ∧ Y_{0} ∧ ⋯ ∧ Ŷ_{j} ∧ ⋯ ∧ Y_{n} ⊗ (v ∧ Y_{j}) ⊲ θ^{i} \\ + ∑_{i,j,k}(−1)^{j+k+1}[Y_{j}, Y_{k}] ∧ X_{i} ∧ Y_{0} ∧ ⋯ ∧ Ŷ_{j} ∧ ⋯ ∧ Ŷ_{k} ∧ ⋯ ∧ Y_{n} ⊗ v ⊲ θ^{i}. \end{matrix} (4.7)

Therefore, the complex is a mixed complex if and only if

\begin{matrix}(∂_{CE} ∘ ∂_{K} + ∂_{K} ∘ ∂_{CE})(Y_{0} ∧ ⋯ ∧ Y_{n} ⊗ v) = \\ ∑_{i} Y_{0} ∧ ⋯ ∧ Y_{n} ⊗ (v ⊲ θ^{i}) · X_{i}+ \\ ∑_{i,j}(−1)^{j+1}X_{i} ∧ Y_{0} ∧ ⋯ ∧ Ŷ_{j} ∧ ⋯ ∧ Y_{n} ⊗ [(v ⊲ θ^{i}) · Y_{j} − (v · Y_{j}) ⊲ θ^{i}]+ \\ ∑_{i,j}(−1)^{j+1}[X_{i}, Y_{j}] ∧ Y_{0} ∧ ⋯ ∧ Ŷ_{j} ∧ ⋯ ∧ Y_{n} ⊗ v ⊲ θ^{i} = 0. \end{matrix} (4.8)

Now, if we assume that (C^{•}(𝔤, V), ∂_{CE} + ∂_{K}) is a differential complex, then we obtain the stability condition (4.5) evaluating (4.8) on 1 ⊗ v. Similarly we observe that V is a 𝔤̃-module by evaluating (4.8) on Y ⊗ v. The converse argument is obvious.

Documenta Mathematica 17 (2012) 483–515

Proposition 4.3. A vector space V is a unimodular stable right 𝔤̃-module, if and only if V ⊗ ℂ_{δ} is a stable right 𝔤̃-module.

Proof. Indeed, if V is unimodular stable right 𝔤̃-module, that is ∑_{i}(v⊲X_{i})·θ^{i} = 0, for any v ∈ V, then

\begin{matrix}∑_{i}((v ⊗ 1_{ℂ}) · θ^{i}) ⊲ X^{i} = ∑_{i}(v · θ^{i}) · X_{i} ⊗ 1_{ℂ} + vδ(X_{i}) ⊗ 1_{ℂ} \\ = ∑_{i}(v · X_{i}) ⊲ θ^{i} ⊗ 1_{ℂ} = 0, \end{matrix}  (4.9)

which proves that V ⊗ ℂ_{δ} is stable. Similarly we observe that for 1 ≤ i, j ≤ N,

\begin{matrix}((v ⊗ 1_{ℂ}) · X_{j}) ⊲ θ^{i} = (v · X_{j} ⊗ 1_{ℂ} + vδ(X_{j}) ⊗ 1_{ℂ}) ⊲ θ^{i} = \\ ((v · X_{j}) ⊲ θ^{i} + vδ(X_{j}) ⊲ θ^{i}) ⊗ 1_{ℂ} = \\ (v ⊲ (X_{j} ⊳ θ^{i}) + (v ⊲ θ^{i}) · X_{j} + vδ(X_{j}) ⊲ θ^{i}) ⊗ 1_{ℂ} = \\ (v ⊗ 1_{ℂ}) ⊲ (X_{j} ⊳ θ^{i}) + ((v ⊗ 1_{ℂ}) ⊲ θ^{i}) · X_{j} \end{matrix}  (4.10)

i.e., V ⊗ ℂ_{δ} is a right 𝔤̃-module.

The converse argument is similar.

Let us now briefly recall the Poincaré isomorphism by

𝔇_{P} : ∧^{k}𝔤^{∗} → ∧^{N−k}𝔤,  η ↦ ι(η)ϖ,  (4.11)

where ϖ = X_{1} ∧ ⋯ ∧ X_{N} is the covolume element of 𝔤. By definition ι(θ^{i}) : ∧^{•}𝔤 → ∧^{•−1}𝔤 is given by

⟨ι(θ^{i})ξ, θ^{j_{1}} ∧ ⋯ ∧ θ^{j_{r−1}}⟩ := ⟨ξ, θ^{i} ∧ θ^{j_{1}} ∧ ⋯ ∧ θ^{j_{r−1}}⟩, ξ ∈ ∧^{r}𝔤. (4.12)

Finally, for η = θ^{i_{1}} ∧ ⋯ ∧ θ^{i_{k}}, the interior multiplication ι(η) : ∧^{•}𝔤 → ∧^{•−k}𝔤 is defined by

ι(η) := ι(θ^{i_{k}}) ∘ ⋯ ∘ ι(θ^{i_{1}}).  (4.13)

Proposition 4.4. Let V be a stable right 𝔤̃-module. Then the Poincaré isomorphism induces a map of complexes between the complex W (𝔤, V ⊗ℂ_{−δ}) and the complex C(𝔤, V).

Proof. Let us first introduce the notation Ṽ := V ⊗ ℂ_{−δ}. We can identify Ṽ with V as a vector space, but with the right 𝔤-module structure deformed as v ⊲ X := v · X − vδ(X).

We prove the commutativity of the (co)boundaries via the (inverse) Poincaré isomorphism, i.e.,

\begin{matrix}𝔇_{P}^{−1} : ∧^{p}𝔤 ⊗ V → ∧^{N−p}𝔤^{∗} ⊗ Ṽ \\ ξ ⊗ v ↦ 𝔇_{P}^{−1}(ξ ⊗ v), \end{matrix}  (4.14)

where for an arbitrary η ∈ ∧^{N−p}𝔤

⟨η, 𝔇_{P}^{−1}(ξ ⊗ v)⟩ := ⟨ηξ, ω^{∗}⟩v.  (4.15)

Here, ω^{∗} ∈ ∧^{N}𝔤^{∗} is the volume form.

The commutativity of the diagram

\xymatrix{∧^{p}𝔤 ⊗ V \ar[d]_{𝔇_{P}^{−1}} \ar[r]^{∂_{CE}} & ∧^{p−1}𝔤 ⊗ V \ar[d]^{𝔇_{P}^{−1}} \\ ∧^{N−p}𝔤^{∗} ⊗ Ṽ \ar[r]_{d_{CE}} & ∧^{N−p+1}𝔤^{∗} ⊗ Ṽ}

follows from the Poincaré duality in Lie algebra homology - cohomology, [13, Chapter VI, Section 3]. For the commutativity of the diagram

\xymatrix{∧^{p}𝔤 ⊗ V \ar[d]_{𝔇_{P}^{−1}} \ar[r]^{∂_{K}} & ∧^{p+1}𝔤 ⊗ V \ar[d]^{𝔇_{P}^{−1}} \\ ∧^{N−p}𝔤^{∗} ⊗ Ṽ \ar[r]_{d_{K}} & ∧^{N−p−1}𝔤^{∗} ⊗ Ṽ}

we take an arbitrary ξ ∈ ∧^{p}𝔤, η ∈ ∧^{N−p−1}𝔤 and v ∈ V. Then

\begin{matrix}𝔇_{P}^{−1}(∂_{K}(ξ ⊗ v))(η) = ⟨ηX_{i}ξ, ω^{∗}⟩v ⊲ θ^{i} = \\ (−1)^{N−p−1}⟨X_{i}ηξ, ω^{∗}⟩v ⊲ θ^{i} = (−1)^{N−p−1}d_{K}(𝔇_{P}^{−1}(ξ ⊗ v))(η). \end{matrix}  (4.16)

5 Lie algebra coaction and SAYD coefficients

In this section we identify the coefficients we discussed in the previous sections of this paper with stable-anti-Yetter-Drinfeld module over the universal enveloping algebra of the Lie algebra in question. To this end, we introduce the notion of comodule over a Lie algebra.

5.1 SAYD modules and cyclic cohomology of Hopf algebras

Let H be a Hopf algebra. By definition, a character δ : H → ℂ is an algebra map. A group-like σ ∈ H is the dual object of the character, i.e., Δ(σ) = σ⊗σ. The pair (δ, σ) is called a modular pair in involution [6] if

δ(σ) = 1, and S_{δ}^{2} = Ad_{σ},  (5.1)

where Ad_{σ}(h) = σhσ^{−1} and S_{δ} is defined by

S_{δ}(h) = δ(h_{(1)})S(h_{(2)}).  (5.2)

We recall from [10] the definition of a right-left stable-anti-Yetter-Drinfeld module over a Hopf algebra H. Let V be a right module and left comodule over a Hopf algebra H. We say that it is stable-anti-Yetter-Drinfeld (SAYD) module over H if

▼(v · h) = S(h_{(3)})v_{⟨−1⟩}h_{(1)} ⊗ v_{⟨0⟩} · h_{(2)},  v_{⟨0⟩} · v_{⟨−1⟩} = v,  (5.3)

for any v ∈ V and h ∈ H. It is shown in [10] that any MPI defines a one dimensional SAYD module and all one dimensional SAYD modules come this way.

Let V be a right-left SAYD module over a Hopf algebra H. Let

C^{q}(H, V) := V ⊗ H^{⊗q}, q ≥ 0.  (5.4)

We recall

face operators ∂_{i} : C^{q}(H, V) → C^{q+1}(H, V),  0 ≤ i ≤ q + 1
degeneracy operators σ_{j} : C^{q}(H, V) → C^{q−1}(H, V),  0 ≤ j ≤ q − 1

cyclic operators τ : C^{q}(H, V) → C^{q}(H, V),

by

\begin{matrix}∂_{0}(v ⊗ h^{1} ⊗ … ⊗ h^{q}) = v ⊗ 1 ⊗ h^{1} ⊗ … ⊗ h^{q}, \\ ∂_{i}(v ⊗ h^{1} ⊗ … ⊗ h^{q}) = v ⊗ h^{1} ⊗ … ⊗ h^{i}_{(1)} ⊗ h^{i}_{(2)} ⊗ … ⊗ h^{q}, \\ ∂_{q+1}(v ⊗ h^{1} ⊗ … ⊗ h^{q}) = v_{⟨0⟩} ⊗ h^{1} ⊗ … ⊗ h^{q} ⊗ v_{⟨−1⟩}, \\ σ_{j}(v ⊗ h^{1} ⊗ … ⊗ h^{q}) = (v ⊗ h^{1} ⊗ … ⊗ ε(h^{j+1}) ⊗ … ⊗ h^{q}), \\ τ(v ⊗ h^{1} ⊗ … ⊗ h^{q}) = v_{⟨0⟩}h^{1}_{(1)} ⊗ S(h^{1}_{(2)}) · (h^{2} ⊗ … ⊗ h^{q} ⊗ v_{⟨−1⟩}), \end{matrix} (5.5)

where H acts on H^{⊗q} diagonally.

The graded module C^{•}(H, V) endowed with the above operators is then a cocyclic module [9], which means that ∂_{i}, σ_{j} and τ satisfy the following identities

∂_{j}∂_{i} = ∂_{i}∂_{j−1},  if  i < j,
σ_{j}σ_{i} = σ_{i}σ_{j+1},  if  i ≤ j,

σ_{j}∂_{i} = \begin{cases}∂_{i}σ_{j−1}, & if  i < j \\ Id & if  i = j or i = j + 1 \\ ∂_{i−1}σ_{j} & if  i > j + 1, \end{cases}

τ∂_{i} = ∂_{i−1}τ, 1 ≤ i ≤ q
τ∂_{0} = ∂_{q+1},  τσ_{i} = σ_{i−1}τ,  1 ≤ i ≤ q
τσ_{0} = σ_{n}τ^{2}, τ^{q+1} = Id .

(5.6)

We use the face operators to define the Hochschild coboundary

b : C^{q}(H, V) → C^{q+1}(H, V),  b := ∑_{i=0}^{q+1}(−1)^{i}∂_{i}  (5.7)

It is known that b^{2} = 0. As a result, one obtains the Hochschild complex of the coalgebra H with coefficients in the bicomodule V. Here, the right comodule defined trivially. The cohomology of (C^{•}(H, V), b) is denoted by H_{coalg}^{•}(H, V).

We use the rest of the operators to define the Connes boundary operator,

B : C^{q}(H, V) → C^{q−1}(H, V),  B := (∑_{i=0}^{q}(−1)^{qi}τ^{i}) σ_{q−1}τ.  (5.8)

It is shown in [5] that for any cocyclic module we have b^{2} = B^{2} = (b+B)^{2} = 0. As a result, we define the cyclic cohomology of H with coefficients in the SAYD module V, which is denoted by HC^{•}(H, V), as the total cohomology of the bicomplex

C^{p,q}(H, V) = \begin{cases}V ⊗ H^{⊗q−p}, & if 0 ≤ p ≤ q, \\ 0, & otherwise. \end{cases}  (5.9)

We can also define the periodic cyclic cohomology of H with coefficients in V, which is denoted by HP^{∗}(H, V), as the total cohomology of direct sum total of the bicomplex

C^{p,q}(H, V) = \begin{cases}V ⊗ H^{⊗q−p}, & if p ≤ q, \\ 0, & otherwise. \end{cases}  (5.10)

It can be seen that the periodic cyclic complex and hence the cohomology is ℤ_{2} graded.

5.2 SAYD modules over Lie algebras

We need to define the notion of comodule over a Lie algebra 𝔤 to be able to make a passage from the stable 𝔤̃-modules we already defined in the previous sections to SAYD modules over the universal enveloping algebra U(𝔤).

Definition 5.1. We say a vector space V is a left comodule over the Lie algebra 𝔤 if there is a map ▼_{𝔤} : V → 𝔤 ⊗ V such that

v_{[−2]} ∧ v_{[−1]} ⊗ v_{[0]} = 0,  (5.11)

where ▼_{𝔤}(v) = v_{[−1]} ⊗ v_{[0]}, and

v_{[−2]} ⊗ v_{[−1]} ⊗ v_{[0]} = v_{[−1]} ⊗ (v_{[0]})_{[−1]} ⊗ (v_{[0]})_{[0]}.

Proposition 5.2. Let 𝔤 be a Lie algebra and V be a vector space. Then, V is a right S(𝔤^{∗})-module if and only if it is a left 𝔤-comodule.

Proof. Assume that V is a right module over the symmetric algebra S(𝔤^{∗}). Then for any v ∈ V there is an element v_{[−1]} ⊗ v_{[0]} ∈ 𝔤^{∗∗} ⊗ V ≅ 𝔤 ⊗ V such that for any θ ∈ 𝔤^{∗}

v ⊲ θ = v_{[−1]}(θ)v_{[0]} = θ(v_{[−1]})v_{[0]}.  (5.12)

Hence define the linear map ▼_{𝔤} : V → 𝔤 ⊗ V by

v ↦ v_{[−1]} ⊗ v_{[0]}.  (5.13)

The compatibility needed for V to be a right module over S(𝔤^{∗}),which is (v ⊲ θ) ⊲ η − (v ⊲ η) ⊲ θ = 0 translates directly into

α(v_{[−2]} ∧ v_{[−1]}) ⊗ v_{[0]} = (v_{[−2]} ⊗ v_{[−1]} − v_{[−1]} ⊗ v_{[−2]}) ⊗ v_{[0]} = 0,  (5.14)

where α : ∧^{2}𝔤 → U(𝔤)^{⊗ 2} is the anti-symmetrization map. Since the antisymmetrization is injective, we have

v_{[−2]} ∧ v_{[−1]} ⊗ v_{[0]} = 0.  (5.15)

Hence, V is a left 𝔤-comodule.

Conversely, assume that V is a left 𝔤-comodule via the map ▼_{𝔤} : V → 𝔤 ⊗ V defined by v ↦ v_{[−1]} ⊗ v_{[0]}. We define the right action

V ⊗ S(𝔤^{∗}) → V, v ⊗ θ ↦ v ⊲ θ := θ(v_{[−1]})v_{[0]},  (5.16)

for any θ ∈ 𝔤^{∗} and any v ∈ V. Thus,

(v ⊲ θ) ⊲ η − (v ⊲ η) ⊲ θ = (v_{[−2]} ⊗ v_{[−1]} − v_{[−1]} ⊗ v_{[−2]})(θ ⊗ η) ⊗ v_{[0]} = 0, (5.17)

proving that V is a right module over S(𝔤^{∗}).

Having understood the relation between the left 𝔤-coaction and right S(𝔤^{∗})- action, it is natural to investigate the relation with left U(𝔤)-coaction.

Let ▼ : V → U(𝔤) ⊗ V be a left U(𝔤)-comodule structure on the linear space V. Then composing via the canonical projection π : U(𝔤) → 𝔤, we get a linear map ▼_{𝔤} : V → 𝔤 ⊗ V.

\xymatrix{V \ar[dr]_{▼_{𝔤}} \ar[r]^{▼} & U(𝔤) ⊗ V \ar[d]^{π⊗id} \\  & 𝔤 ⊗ V}

Lemma 5.3. If ▼ : V → U(𝔤) ⊗ V is a coaction, then so is ▼_{𝔤} : V → 𝔤 ⊗ V.

Proof. If we write ▼(v) = v_{[\overline{−1}]} ⊗ v_{[\overline{0}]} then

\begin{matrix}v_{[−2]} ∧ v_{[−1]} ⊗ v_{[0]} = π(v_{[\overline{−2}]}) ∧ π(v_{[\overline{−1}]}) ⊗ v_{[\overline{0}]} = \\ π(v_{[\overline{−1}]}_{(1)}) ∧ π(v_{[\overline{−1}]}_{(2)}) ⊗ v_{[\overline{0}]} = 0 \end{matrix}  (5.18)

by the cocommutativity of U(𝔤).

For the reverse process which is to obtain a U(𝔤)-comodule out of a 𝔤-comodule, we will need the following concept.

Definition 5.4. Let V be a 𝔤-comodule via ▼_{𝔤} : V → 𝔤 ⊗ V. Then we call the coaction locally conilpotent if it is conilpotent on any one dimensional subspace. In other words, ▼_{𝔤} : V → 𝔤 ⊗ V is locally conilpotent if and only if for any v ∈ V there exists n ∈ ℕ such that ▼_{𝔤}^{n}(v) = 0.

Example 5.5. If V is an SAYD module on U(𝔤), then by [11, Lemma 6.2] we have the filtration V = ∪_{p∈ℤ}F_{p}V defined as F_{0}V = V^{coU(𝔤)} and inductively

F_{p+1}V/F_{p}V = (V/F_{p}V)^{coU(𝔤)}  (5.19)

Then the induced 𝔤-comodule V is locally conilpotent.

Example 5.6. Let 𝔤 be a Lie algebra and S(𝔤^{∗}) be the symmetric algebra on 𝔤^{∗}. For V = S(𝔤^{∗}), consider the coaction

S(𝔤^{∗}) → 𝔤 ⊗ S(𝔤^{∗}), α ↦ X_{i} ⊗ αθ^{i},  (5.20)

called the Koszul coaction. The corresponding S(𝔤^{∗})-action on V coincides with the multiplication of S(𝔤^{∗}). Therefore, the Koszul coaction is not locally conilpotent.

One notes that the Koszul coaction is locally conilpotent on any truncation of the symmetric algebra.

Let {U_{k}(𝔤)}_{k≥0} be the canonical filtration of U(𝔤), i.e.,

U_{0}(𝔤) = ℂ · 1, U_{1}(𝔤) = ℂ · 1 ⊕ 𝔤, U_{p}(𝔤) · U_{q}(𝔤) ⊆ U_{p+q}(𝔤)  (5.21)

Let us call an element in U(𝔤) as symmetric homogeneous of degree k if it is the canonical image of a symmetric homogeneous tensor of degree k over 𝔤. Let U^{k}(𝔤) be the set of all symmetric elements of degree n in U(𝔤).

We recall from [7, Proposition 2.4.4] that

U_{k}(𝔤) = U_{k−1}(𝔤) ⊕ U^{k}(𝔤).  (5.22)

In other words, there is a (canonical) projection

\begin{matrix}θ_{k} : U_{k}(𝔤) → U^{k}(𝔤) ≅ U_{k}(𝔤)/U_{k−1}(𝔤) \\ X_{1} ⋯ X_{k} ↦ ∑_{σ∈S_{k}} X_{σ(1)} ⋯ X_{σ(k)}. \end{matrix}  (5.23)

So, fixing an ordered basis of the Lie algebra 𝔤, we can say that the above map is bijective on the PBW-basis elements.

Let us consider the unique derivation of U(𝔤) extending the adjoint action of the Lie algebra 𝔤 on itself, and call it ad(X) : U(𝔤) → U(𝔤) for any X ∈ 𝔤. By [7, Proposition 2.4.9], ad(X)(U^{k}(𝔤)) ⊆ U^{k}(𝔤) and ad(X)(U_{k}(𝔤)) ⊆ U_{k}(𝔤). So by applying ad(X) to both sides of (5.22), we observe that the preimage of ad(Y)(∑_{σ∈S_{k}} X_{σ(1)} ⋯ X_{σ(k)}) is ad(Y)(X_{1} ⋯ X_{k}).

Proposition 5.7. For a locally conilpotent 𝔤-comodule V, the linear map

\begin{matrix}▼ : V → U(𝔤) ⊗ V \\ v ↦ 1 ⊗ v + ∑_{k≥1} θ_{k}^{−1}(v_{[−k]} ⋯ v_{[−1]}) ⊗ v_{[0]} \end{matrix}  (5.24)

defines a U(𝔤)-comodule structure.

Proof. For an arbitrary basis element v^{i} ∈ V, let us write

v^{i}_{[−1]} ⊗ v^{i}_{[0]} = α_{k}^{ij}X_{j} ⊗ v^{k}  (5.25)

where α_{k}^{ij} ∈ ℂ. Then, by the coaction compatibility v_{[−2]} ∧ v_{[−1]} ⊗ v_{[0]} = 0 we have

v^{i}_{[−2]} ⊗ v^{i}_{[−1]} ⊗ v^{i}_{[0]} = ∑_{j_{1},j_{2}} α_{l_{2}}^{ij_{1}j_{2}}X_{j_{1}} ⊗ X_{j_{2}} ⊗ v^{l_{2}},  (5.26)

such that α_{l_{2}}^{ij_{1}j_{2}} := α_{l_{1}}^{ij_{1}}α_{l_{2}}^{l_{1}j_{2}} and α_{l_{2}}^{ij_{1}j_{2}} = α_{l_{2}}^{ij_{2}j_{1}}.

We have

▼(v^{i}) = 1 ⊗ v^{i} + ∑_{k≥1} ∑_{j_{1}≤···≤j_{k}} α_{l_{k}}^{ij_{1}···j_{k}}X_{j_{1}} ⋯ X_{j_{k}} ⊗ v^{l_{k}},  (5.27)

because for k ≥ 1

v^{i}_{[−k]} ⊗ ⋯ ⊗ v^{i}_{[−1]} ⊗ v^{i}_{[0]} = ∑_{j_{1},··· ,j_{k}} α_{l_{k}}^{ij_{1}···j_{k}}X_{j_{1}} ⊗ ⋯ ⊗ X_{j_{k}} ⊗ v^{l_{k}}, (5.28)

where α_{l_{k}}^{ij_{1}···j_{k}} := α_{l_{1}}^{ij_{1}} ⋯ α_{l_{k}}^{l_{k−1}j_{k}}, and for any σ ∈ S_{k} we have

α_{l_{k}}^{ij_{1}···j_{k}} = α_{l_{k}}^{ij_{σ(1)}···j_{σ(k)}}.  (5.29)

At this point, the counitality is immediate,

(ε ⊗ id) ∘ ▼(v^{i}) = v^{i}.  (5.30)

On the other hand, to prove the coassociativity we first observe that

\begin{matrix}(id ⊗ ▼) ∘ ▼(v^{i}) = 1 ⊗ ▼(v^{i}) + ∑_{k≥1} ∑_{j_{1}≤···≤j_{k}} α_{l_{k}}^{ij_{1}···j_{k}}X_{j_{1}} ⋯ X_{j_{k}} ⊗ ▼(v^{l_{k}}) \\ = 1 ⊗ 1 ⊗ v^{i} + ∑_{k≥1} ∑_{j_{1}≤···≤j_{k}} α_{l_{k}}^{ij_{1}···j_{k}}1 ⊗ X_{j_{1}} ⋯ X_{j_{k}} ⊗ v^{l_{k}}+ \\ ∑_{k≥1} ∑_{j_{1}≤···≤j_{k}} α_{l_{k}}^{ij_{1}···j_{k}}X_{j_{1}} ⋯ X_{j_{k}} ⊗ 1 ⊗ v^{l_{k}}+ \\ ∑_{k≥1} ∑_{j_{1}≤···≤j_{k}} α_{l_{k}}^{ij_{1}···j_{k}}X_{j_{1}} ⋯ X_{j_{k}} ⊗ (∑_{t≥1} ∑_{r_{1}≤···≤r_{t}} α_{s_{t}}^{l_{k}r_{1}···r_{t}}X_{r_{1}} ⋯ X_{r_{t}} ⊗ v^{s_{t}}), \end{matrix} (5.31)

where α_{s_{t}}^{l_{k}r_{1}···r_{t}} := α_{s_{1}}^{l_{k}r_{1}} ⋯ α_{s_{t}}^{s_{t−1}r_{t}}. Then we notice that

\begin{matrix}Δ(∑_{k≥1} ∑_{j_{1}≤···≤j_{k}} α_{l_{k}}^{ij_{1}···j_{k}}X_{j_{1}} ⋯ X_{j_{k}}) ⊗ v^{l_{k}} \\ = ∑_{k≥1} ∑_{j_{1}≤···≤j_{k}} α_{l_{k}}^{ij_{1}···j_{k}}1 ⊗ X_{j_{1}} ⋯ X_{j_{k}} ⊗ v^{l_{k}} \\ + ∑_{k≥1} ∑_{j_{1}≤···≤j_{k}} α_{l_{k}}^{ij_{1}···j_{k}}X_{j_{1}} ⋯ X_{j_{k}} ⊗ 1 ⊗ v^{l_{k}} \\ + ∑_{k≥2} ∑_{j_{1}≤···≤r_{1}≤···≤r_{p}≤···≤j_{k}} α_{l_{k}}^{ij_{1}···j_{k}}X_{r_{1}} ⋯ X_{r_{p}} ⊗ X_{j_{1}} ⋯ X̂_{r_{1}} ⋯ X̂_{r_{p}} ⋯ X_{j_{k}} ⊗ v^{l_{k}} \\ = ∑_{k≥1} ∑_{j_{1}≤···≤j_{k}} α_{l_{k}}^{ij_{1}···j_{k}}1 ⊗ X_{j_{1}} ⋯ X_{j_{k}} ⊗ v^{l_{k}} \\ + ∑_{k≥1} ∑_{j_{1}≤···≤j_{k}} α_{l_{k}}^{ij_{1}···j_{k}}X_{j_{1}} ⋯ X_{j_{k}} ⊗ 1 ⊗ v^{l_{k}} \\ + ∑_{p≥1} ∑_{k−p≥1} \\ ∑_{q_{1}≤···≤q_{k−p}} ∑_{r_{1}≤···≤r_{p}} α_{l_{p}}^{ir_{1}···r_{p}}α_{l_{k}}^{l_{p}q_{1}···q_{k−p}}X_{r_{1}} ⋯ X_{r_{p}} ⊗ X_{q_{1}} ⋯ X_{q_{k−p}} ⊗ v^{l_{k}}, \end{matrix} (5.32)

where for the last equality we write the complement of r_{1} ≤ ⋯ ≤ r_{p} in j_{1} ≤ ⋯ ≤ j_{k} as q_{1} ≤ ⋯ ≤ q_{k−p}. Then (5.29) implies that

α_{l_{k}}^{ij_{1}···j_{k}} = α_{l_{k}}^{ir_{1}···r_{p}q_{1}···q_{k−p}} = α_{l_{p}}^{ir_{1}···r_{p}}α_{l_{k}}^{l_{p}q_{1}···q_{k−p}}.  (5.33)

As a result,

(id ⊗ ▼) ∘ ▼(v^{i}) = (Δ ⊗ id) ∘ ▼(v^{i}).  (5.34)

This is the coassociativity and the proof is now complete.

Documenta Mathematica 17 (2012) 483–515

Let us denote by ^{𝔤}conilM the subcategory of locally conilpotent left 𝔤comodules of the category of left 𝔤-comodules ^{𝔤}M with colinear maps.

Assigning a 𝔤-comodule ▼_{𝔤} : V → 𝔤⊗V to a U(𝔤)-comodule ▼ : V → U(𝔤)⊗V determines a functor

\xymatrix{^{U(𝔤)}M \ar[r]^{P} & ^{𝔤}conilM}  (5.35)

Similarly, constructing a U(𝔤)-comodule from a 𝔤-comodule determines a functor

\xymatrix{^{𝔤}conilM \ar[r]^{E} & ^{U(𝔤)}M}  (5.36)

As a result, we can express the following proposition.

Proposition 5.8. The categories ^{U(𝔤)}M and ^{𝔤}conilM are isomorphic.

Proof. We show that the functors

\xymatrix{^{U(𝔤)}M \ar[r]^{P} & ^{𝔤}conilM \ar[l]^{E}}

are inverses to each other.

If ▼_{𝔤} : V → 𝔤 ⊗ V is a locally conilpotent 𝔤-comodule and ▼ : V → U(𝔤) ⊗ V the corresponding U(𝔤)-comodule, by the very definition the 𝔤-comodule corresponding to ▼ : V → U(𝔤) ⊗ V is exactly ▼_{𝔤} : V → 𝔤 ⊗ V. This proves that

P ∘ E = Id_{^{𝔤}conilM} .  (5.37)

Conversely, let us start with a U(𝔤)-comodule ▼ : V → U(𝔤) ⊗ V and write the coaction by using the PBW-basis of U(𝔤) as follows

v^{i}_{(−1)} ⊗ v^{i}_{(0)} = 1 ⊗ v^{i} + ∑_{k≥1} ∑_{j_{1}≤···≤j_{k}} γ_{l_{k}}^{ij_{1}···j_{k}}X_{j_{1}} ⋯ X_{j_{k}} ⊗ v^{l_{k}}.  (5.38)

So, the corresponding 𝔤-comodule ▼_{𝔤} : V → 𝔤 ⊗ V is given as follows

v^{i}_{[−1]} ⊗ v^{i}_{[0]} = π(v^{i}_{(−1)}) ⊗ v^{i}_{(0)} = ∑_{j} γ_{k}^{ij}X_{j} ⊗ v^{k}.  (5.39)

Finally, the U(𝔤)-coaction corresponding to this 𝔤-coaction is defined on v^{i} ∈ V as

v^{i} ↦ 1 ⊗ v + ∑_{k≥1} ∑_{j_{1}≤···≤j_{k}} γ_{l_{1}}^{ij_{1}}γ_{l_{2}}^{l_{1}j_{2}} ⋯ γ_{l_{k}}^{l_{k−1}j_{k}}X_{j_{1}} ⋯ X_{j_{k}} ⊗ v^{l_{k}}  (5.40)

Therefore, we can recover U(𝔤)-coaction we started with if and only if

γ_{l_{k}}^{ij_{1}···j_{k}} = γ_{l_{1}}^{ij_{1}}γ_{l_{2}}^{l_{1}j_{2}} ⋯ γ_{l_{k}}^{l_{k−1}j_{k}}, ∀k ≥ 1.  (5.41)

The equation (5.41) is a consequence of the coassociativity ▼. Indeed, applying the coassociativity as

(Δ^{k−1} ⊗ id) ∘ ▼ = ▼^{k}  (5.42)

and comparing the coefficients of X_{j_{1}} ⊗ ⋯ ⊗ X_{j_{k}} we conclude (5.41) for any k ≥ 1. Hence, we proved

E ∘ P = Id_{^{U(𝔤)}M} .  (5.43)

The equation (5.41) implies that if ▼ : V → U(𝔤) ⊗ V is a left coaction, then its associated 𝔤-coaction ▼_{𝔤} : V → 𝔤 ⊗ V is locally conilpotent.

For a 𝔤-coaction

v ↦ v_{[−1]} ⊗ v_{[0]},  (5.44)

the associated U(𝔤)-coaction is denoted by

v ↦ v_{[\overline{−1}]} ⊗ v_{[\overline{0}]}.  (5.45)

Definition 5.9. Let V be a right module and left comodule over a Lie algebra 𝔤. We call V a right-left AYD over 𝔤 if

▼_{𝔤}(v · X) = v_{[−1]} ⊗ v_{[0]} · X + [v_{[−1]}, X] ⊗ v_{[0]}.  (5.46)

Moreover, V is called stable if

v_{[0]} · v_{[−1]} = 0.  (5.47)

Proposition 5.10. Let ▼_{𝔤} : V → 𝔤 ⊗ V be a locally conilpotent 𝔤-comodule and ▼ : V → U(𝔤) ⊗ V the corresponding U(𝔤)-comodule structure. Then, V is a right-left AYD over 𝔤 if and only if it is a right-left AYD over U(𝔤).

Proof. Let us first assume V to be a right-left AYD module over 𝔤. For X ∈ 𝔤 and an element v ∈ V, AYD compatibility implies that

\begin{matrix}(v · X)_{[−k]} ⊗ ⋯ ⊗ (v · X)_{[−1]} ⊗ (v · X)_{[0]} = v_{[−k]} ⊗ ⋯ ⊗ v_{[−1]} ⊗ v_{[0]} · X \\ + [v_{[−k]}, X] ⊗ ⋯ ⊗ v_{[−1]} ⊗ v_{[0]} + v_{[−k]} ⊗ ⋯ ⊗ [v_{[−1]}, X] ⊗ v_{[0]}. \end{matrix} (5.48)

Multiplying in U(𝔤), we get

\begin{matrix}(v · X)_{[−k]} ⋯ (v · X)_{[−1]} ⊗ (v · X)_{[0]} = \\ v_{[−k]} ⋯ v_{[−1]} ⊗ v_{[0]} · X − ad(X)(v_{[−k]} ⋯ v_{[−1]}) ⊗ v_{[0]}. \end{matrix}  (5.49)

So, for the extension ▼ : V → U(𝔤) ⊗ V we have

\begin{matrix}(v · X)_{[\overline{−1}]} ⊗ (v · X)_{[\overline{0}]} = 1 ⊗ v · X + ∑_{k≥1} θ_{k}^{−1}((v · X)_{[−k]} ⋯ (v · X)_{[−1]}) ⊗ (v · X)_{[0]} \\ = 1 ⊗ v · X + ∑_{k≥1} θ_{k}^{−1}(v_{[−k]} ⋯ v_{[−1]}) ⊗ v_{[0]} · X − ∑_{k≥1} θ_{k}^{−1}(ad(X)(v_{[−k]} ⋯ v_{[−1]})) ⊗ v_{[0]} \\ = v_{[\overline{−1}]} ⊗ v_{[\overline{0}]} · X − ∑_{k≥1} ad(X)(θ_{k}^{−1}(v_{[−k]} ⋯ v_{[−1]})) ⊗ v_{[0]} \\ = v_{[\overline{−1}]} ⊗ v_{[\overline{0}]} · X − ad(X)(v_{[\overline{−1}]}) ⊗ v_{[\overline{0}]} = S(X_{(3)})v_{[\overline{−1}]}X_{(1)} ⊗ v_{[\overline{0}]} · X_{(2)}. \end{matrix} (5.50)

Here on the third equality we used the fact that the operator ad commute with θ_{k}, and on the fourth equality we used

\begin{matrix}∑_{k≥1} ad(X)(θ_{k}^{−1}(v_{[−k]} ⋯ v_{[−1]})) ⊗ v_{[0]} = \\ ∑_{k≥1} ad(X)(θ_{k}^{−1}(v_{[−k]} ⋯ v_{[−1]})) ⊗ v_{[0]} + ad(X)(1) ⊗ v = ad(X)(v_{[\overline{−1}]}) ⊗ v_{[\overline{0}]}. \end{matrix} (5.51)

By using the fact that AYD condition is multiplicative, we conclude that ▼ : M → U(𝔤) ⊗ M satisfies the AYD condition on U(𝔤).

Conversely assume that V is a right-left AYD over U(𝔤). We first observe that

(Δ ⊗ id) ∘ Δ(X) = X ⊗ 1 ⊗ 1 + 1 ⊗ X ⊗ 1 + 1 ⊗ 1 ⊗ X  (5.52)

Accordingly,

\begin{matrix}▼(v · X) = v_{[\overline{−1}]}X ⊗ v_{[\overline{0}]} + v_{[\overline{−1}]} ⊗ v_{[\overline{0}]} · X − Xv_{[\overline{−1}]} ⊗ v_{[\overline{0}]} \\ = −ad(X)(v_{[\overline{−1}]}) ⊗ v_{[\overline{0}]} + v_{[\overline{−1}]} ⊗ v_{[\overline{0}]} · X \end{matrix}  (5.53)

It is known that the projection map π : U(𝔤) → 𝔤 commutes with the adjoint representation. So

\begin{matrix}▼_{𝔤}(v · X) = −π(ad(X)(v_{[\overline{−1}]})) ⊗ v_{[\overline{0}]} + π(v_{[\overline{−1}]}) ⊗ v_{[\overline{0}]} · X \\ = −ad(X)π(v_{[\overline{−1}]}) ⊗ v_{[\overline{0}]} + π(v_{[\overline{−1}]}) ⊗ v_{[\overline{0}]} · X \\ = [v_{[−1]}, X] ⊗ v_{[0]} + v_{[−1]} ⊗ v_{[0]} · X. \end{matrix}  (5.54)

That is, V is a right-left AYD over 𝔤.

Lemma 5.11. Let ▼_{𝔤} : V → 𝔤 ⊗ V be a locally conilpotent 𝔤-comodule and ▼ : V → U(𝔤) ⊗ V be the corresponding U(𝔤)-comodule structure. If V is stable over 𝔤, then it is stable over U(𝔤).

Proof. Writing the 𝔤-coaction in terms of basis elements as in (5.25), the stability reads

v^{i}_{[0]}v^{i}_{[−1]} = α_{k}^{ij}v^{k} · X_{j} = 0, ∀i  (5.55)

Therefore, for the corresponding U(𝔤)-coaction we have

\begin{matrix}∑_{j_{1}≤···≤j_{k}} α_{l_{1}}^{ij_{1}} ⋯ α_{l_{k}}^{l_{k−1}j_{k}}v^{l_{k}} · (X_{j_{1}} ⋯ X_{j_{k}}) = \\ ∑_{j_{1}≤···≤j_{k−1}} α_{l_{1}}^{ij_{1}} ⋯ α_{l_{k−1}}^{l_{k−2}j_{k−1}}(∑_{j_{k}} α_{l_{k}}^{l_{k−1}j_{k}}v^{l_{k}} · X_{j_{1}}) · (X_{j_{2}} ⋯ X_{j_{k}}) = \\ ∑_{j_{2},··· ,j_{k}} α_{l_{1}}^{ij_{k}} ⋯ α_{l_{k−1}}^{l_{k−2}j_{k−1}}(∑_{j_{1}} α_{l_{k}}^{l_{k−1}j_{1}}v^{l_{k}} · X_{j_{1}}) · (X_{j_{2}} ⋯ X_{j_{k}}), \end{matrix} (5.56)

where on the second equality we used (5.29). This immediately implies that

v^{i}_{[\overline{0}]} · v^{i}_{[\overline{−1}]} = v^{i}.  (5.57)

That is, the stability over U(𝔤).

However, the converse is not true.

Example 5.12. It is known that U(𝔤), as a left U(𝔤)-comodule via Δ : U(𝔤) → U(𝔤) ⊗ U(𝔤) and a right 𝔤-module via ad : U(𝔤) ⊗ 𝔤 → U(𝔤) is stable. However, the associated 𝔤-comodule, is no longer stable. Indeed, for u = X_{1}X_{2}X_{3} ∈ U(𝔤), we have

u_{[−1]} ⊗ u_{[0]} = X_{1} ⊗ X_{2}X_{3} + X_{2} ⊗ X_{1}X_{3} + X_{3} ⊗ X_{1}X_{2}  (5.58)

Then,

u_{[0]} · u_{[−1]} = [[X_{1}, X_{2}], X_{3}] + [[X_{2}, X_{1}], X_{3}] + [[X_{1}, X_{3}], X_{2}] = [[X_{1}, X_{3}], X_{2}]

(5.59)

which is not necessarily zero.

The following is the main result of this section.

Proposition 5.13. Let V be a vector space, and 𝔤 be a Lie algebra. Then, V is a stable right 𝔤̃-module if and only if it is a right-left SAYD module over 𝔤.

Proof. Let us first assume that V is a stable right 𝔤̃-module. Since V is a right S(𝔤^{∗})-module it is a left 𝔤-comodule by Proposition 5.2. Accordingly

\begin{matrix}[v_{[−1]}, X_{j}] ⊗ v_{[0]} + v_{[−1]} ⊗ v_{[0]} · X_{j} = \\ [X_{l}, X_{j}]θ^{l}(v_{[−1]}) ⊗ v_{[0]} + X_{t}θ^{t}(v_{[−1]}) ⊗ v_{[0]} · X_{j} = \\ X_{t}C_{lj}^{t}θ^{l}(v_{[−1]}) ⊗ v_{[0]} + X_{t}θ^{t}(v_{[−1]}) ⊗ v_{[0]} · X_{j} = \\ X_{t} ⊗ [v ⊲ (X_{j} ⊳ θ^{t}) + (v ⊲ θ^{t}) · X_{j}] = \\ X_{t} ⊗ (v · X_{j}) ⊲ θ^{t} = X_{t}θ^{t}((v · X_{j})_{[−1]}) ⊗ (v · X_{j})_{[0]} = \\ (v · X_{j})_{[−1]} ⊗ (v · X_{j})_{[0]} \end{matrix}  (5.60)

This proves that V is a right-left AYD module over 𝔤. On the other hand, for any v ∈ V,

v_{[0]} · v_{[−1]} = ∑_{i} v_{[0]} · X_{i}θ^{i}(v_{[−1]}) = ∑_{i}(v ⊲ θ^{i}) · X_{i} = 0  (5.61)

Hence, V is stable too. As a result, V is SAYD over 𝔤.

Conversely, assume that V is a right-left SAYD module over 𝔤. So V is a right module over S(𝔤^{∗}) and a right module over 𝔤. In addition we see that

\begin{matrix}v ⊲ (X_{j} ⊳ θ^{i}) + (v ⊲ θ^{i}) · X_{j} = C_{kj}^{i}v ⊲ θ^{k} + (v ⊲ θ^{i}) · X_{j} = \\ C_{kj}^{i}θ^{k}(v_{[−1]})v_{[0]} + θ^{i}(v_{[−1]})v_{[0]} · X_{j} = \\ θ^{i}([v_{[−1]}, X_{j}])v_{[0]} + θ^{i}(v_{[−1]})v_{[0]} · X_{j} = \\ (θ^{i} ⊗ id)([v_{[−1]}, X_{j}] ⊗ v_{[0]} + v_{[−1]} ⊗ v_{[0]} · X_{j}) = \\ θ^{t}((v · X_{j})_{[−1]})(v · X_{j})_{[0]} = (v · X_{j}) ⊲ θ^{i}. \end{matrix}  (5.62)

Thus, V is a right 𝔤̃-module. Finally, we prove the stability by

∑_{i}(v ⊲ θ^{i}) · X_{i} = ∑_{i} v_{[0]} · X_{i}θ^{i}(v_{[−1]}) = v_{[0]} · v_{[−1]} = 0.  (5.63)

Corollary 5.14. Any right module over the Weyl algebra D(𝔤) is a right-left SAYD module over the Lie algebra 𝔤.

Finally, we state an analogous of Lemma 2.3 [10] to show that the category of 𝔤AYD_{𝔤} is monoidal.

Proposition 5.15. Let M and N be two right-left AYD modules over 𝔤. Then M ⊗ N is also a right-left AYD over 𝔤 via the coaction

▼_{𝔤} : M ⊗ N → 𝔤 ⊗ M ⊗ N, m ⊗ n ↦ m_{[−1]} ⊗ m_{[0]} ⊗ n + n_{[−1]} ⊗ m ⊗ n_{[0]} (5.64)

and the action

M ⊗ N ⊗ 𝔤 → M ⊗ N, (m ⊗ n) · X = m · X ⊗ n + m ⊗ n · X  (5.65)

Proof. We verify that

\begin{matrix}[(m ⊗ n)_{[−1]}, X] ⊗ (m ⊗ n)_{[0]} + (m ⊗ n)_{[−1]} ⊗ (m ⊗ n)_{[0]} · X = \\ [m_{[−1]}, X] ⊗ m_{[0]} ⊗ n + [n_{[−1]}, X] ⊗ m ⊗ n_{[0]}+ \\ m_{[−1]} ⊗ (m_{[0]} ⊗ n) · X + n_{[−1]} ⊗ (m ⊗ n_{[0]}) · X = \\ (m · X)_{[−1]} ⊗ (m · X)_{[0]} ⊗ n + n_{[−1]} ⊗ m · X ⊗ n_{[0]}+ \\ m_{[−1]} ⊗ m_{[0]} ⊗ n · X + (n · X)_{[−1]} ⊗ m ⊗ (n · X)_{[0]} = \\ ▼_{𝔤}(m · X ⊗ n + m ⊗ n · X) = ▼_{𝔤}((m ⊗ n) · X). \end{matrix}  (5.66)

5.3 Examples

This subsection is devoted to examples to illustrate the notion of SAYD module over a Lie algebra. We consider the representations and corepresentations of a Lie algebra 𝔤 on a finite dimensional vector space V in terms of matrices. We then investigate the SAYD condition as a relation between these matrices and the Lie algebra structure of 𝔤.

Let also V be a n dimensional 𝔤-module with a basis {v^{1}, ⋯, v^{n}}. We express the module structure as

m^{i} · X_{j} = β_{jk}^{i}m^{k}, β_{jk}^{i} ∈ ℂ.  (5.67)

In this way, for any basis element X_{j} ∈ 𝔤 we obtain a matrix B_{j} ∈ M_{n}(ℂ) such that

(B_{j})_{k}^{i} := β_{jk}^{i}.  (5.68)

Let ▼_{𝔤} : V → 𝔤 ⊗ V be a coaction. We write the coaction as

▼_{𝔤}(v^{i}) = α_{k}^{ij}X_{j} ⊗ v^{k}, α_{k}^{ij} ∈ ℂ.  (5.69)

This way we get a matrix A^{j} ∈ M_{n}(ℂ) for any basis element X_{j} ∈ 𝔤 such that

(A^{j})_{k}^{i} := α_{k}^{ij}.  (5.70)

Lemma 5.16. Linear map ▼_{𝔤} : M → 𝔤 ⊗ M forms a right 𝔤-comodule if and only if

A^{j_{1}} · A^{j_{2}} = A^{j_{2}} · A^{j_{1}}.  (5.71)

Proof. It is just the translation of the coaction compatibility v^{i}_{[−2]}∧v^{i}_{[−1]}⊗v^{i}_{[0]} = 0 in terms of the matrices A^{i}.

Lemma 5.17. Right 𝔤-module left 𝔤-comodule V is stable if and only if

∑_{j} A^{j} · B_{j} = 0.  (5.72)

Proof. By the definition of the stability,

v^{i}_{[0]} · v^{i}_{[−1]} = α_{k}^{ij}v^{k} · X_{j} = α_{k}^{ij}β_{jl}^{k}v^{l} = 0  (5.73)

Therefore,

α_{k}^{ij}β_{jl}^{k} = (A^{j})_{k}^{i}(B_{j})_{l}^{k} = (A^{j} · B_{j})_{l}^{i} = 0.  (5.74)

We proceed to express the AYD condition.

Lemma 5.18. The 𝔤-module-comodule V is a right-left AYD if and only if

[B_{q}, A^{j}] = ∑_{s} A^{s}C_{sq}^{j}.  (5.75)

Proof. We first observe

\begin{matrix}▼_{𝔤}(v^{p} · X_{q}) = ▼_{𝔤}(β_{qk}^{p}v^{k}) = β_{qk}^{p}α_{l}^{kj}X_{j} ⊗ v^{l} \\ = (B_{q})_{k}^{p}(A^{j})_{l}^{k}X_{j} ⊗ v^{l} = (B_{q} · A^{j})_{l}^{p}X_{j} ⊗ v^{l}. \end{matrix}  (5.76)

On the other hand, writing ▼_{𝔤}(v^{p}) = α_{l}^{pj}X_{j} ⊗ v^{l},

\begin{matrix}[v^{p}_{[−1]}, X_{q}] ⊗ v^{p}_{[0]} + v^{p}_{[−1]} ⊗ v^{p}_{[0]} · X_{q} = α_{l}^{ps}[X_{s}, X_{q}] ⊗ v^{l} + α_{t}^{pj}X_{j} ⊗ v^{t} · X_{q} \\ = α_{l}^{ps}C_{sq}^{j}X_{j} ⊗ v^{l} + α_{t}^{pj}β_{ql}^{t}X_{j} ⊗ v^{l} = (α_{l}^{ps}C_{sq}^{j} + (A^{j} · B_{q})_{l}^{p})X_{j} ⊗ v^{l}. \end{matrix} (5.77)

Next, considering the Lie algebra 𝔰𝔩(2), we determine the SAYD modules over simple 𝔰𝔩(2)-modules.

Example 5.19. Let V =< {v^{1}, v^{2}} > be a two dimensional simple 𝔰𝔩(2)- module. Then, by [12], the representation

ρ : 𝔰𝔩(2) → 𝔤𝔩(V)  (5.78)

is the inclusion ρ : 𝔰𝔩(2) ↪ 𝔤𝔩(2). Therefore, we have

B_{1} = ( \begin{matrix}0 & 0 \\ 1 & 0 \end{matrix} ) , B_{2} = ( \begin{matrix}0 & 1 \\ 0 & 0 \end{matrix} ) , B_{3} = ( \begin{matrix}1 & 0 \\ 0 & −1 \end{matrix} ) .  (5.79)

We want to find

A^{1} = ( \begin{matrix}x_{1}^{1} & x_{2}^{1} \\ x_{1}^{2} & x_{2}^{2} \end{matrix} ) ,  A^{2} = ( \begin{matrix}y_{1}^{1} & y_{2}^{1} \\ y_{1}^{2} & y_{2}^{2} \end{matrix} ) ,  A^{3} = ( \begin{matrix}z_{1}^{1} & z_{2}^{1} \\ z_{1}^{2} & z_{2}^{2} \end{matrix} ) , (5.80)

such that together with the 𝔤-coaction ▼_{𝔰𝔩(2)} : V → 𝔰𝔩(2) ⊗ V, defined as v^{i} ↦ (A^{j})_{k}^{i}X_{j} ⊗ v^{k}, V becomes a right-left SAYD over 𝔰𝔩(2). We first express the stability condition. To this end,

A^{1}·B_{1} = ( \begin{matrix}x_{2}^{1} & 0 \\ x_{2}^{2} & 0 \end{matrix} ) ,  A^{2}·B_{2} = ( \begin{matrix}0 & y_{1}^{1} \\ 0 & y_{1}^{2} \end{matrix} ) ,  A^{3}·B_{3} = ( \begin{matrix}z_{1}^{1} & −z_{2}^{1} \\ z_{1}^{2} & −z_{2}^{2} \end{matrix} ) ,

(5.81)

and hence, the stability is

∑_{j} A^{j} · B_{j} = ( \begin{matrix}x_{2}^{1} + z_{1}^{1} & y_{1}^{1} − z_{2}^{1} \\ x_{2}^{2} + z_{1}^{2} & y_{1}^{2} − z_{2}^{2} \end{matrix} ) = 0.  (5.82)

Next, we consider the AYD condition

[B_{q}, A^{j}] = ∑_{s} A^{s}C_{sq}^{j}.  (5.83)

For j = 1 = q,

A^{1} = ( \begin{matrix}x_{1}^{1} & 0 \\ x_{1}^{2} & x_{2}^{2} \end{matrix} ) ,  A^{2} = ( \begin{matrix}0 & y_{2}^{1} \\ 0 & y_{2}^{2} \end{matrix} ) ,  A^{3} = ( \begin{matrix}0 & 0 \\ z_{1}^{2} & 0 \end{matrix} ) . (5.84)

Similarly, for q = 2 and j = 1, we arrive

A^{1} = ( \begin{matrix}0 & 0 \\ 0 & 0 \end{matrix} ) ,  A^{2} = ( \begin{matrix}0 & y_{2}^{1} \\ 0 & y_{2}^{2} \end{matrix} ) ,  A^{3} = ( \begin{matrix}0 & 0 \\ 0 & 0 \end{matrix} ) .  (5.85)

Finally, for j = 1 and q = 2 we conclude

A^{1} = ( \begin{matrix}0 & 0 \\ 0 & 0 \end{matrix} ) ,  A^{2} = ( \begin{matrix}0 & 0 \\ 0 & 0 \end{matrix} ) ,  A^{3} = ( \begin{matrix}0 & 0 \\ 0 & 0 \end{matrix} ) .  (5.86)

Thus, the only 𝔰𝔩(2)-comodule structure that makes a 2-dimensional simple 𝔰𝔩(2)-module V to be a right-left SAYD over 𝔰𝔩(2) is the trivial comodule structure.

Example 5.20. We investigate all possible coactions that make the truncated symmetric algebra S(𝔰𝔩(2)^{∗})_{[2]} an SAYD module over 𝔰𝔩(2).

A vector space basis of S(𝔰𝔩(2)^{∗})_{[2]} is {1 = θ^{0}, θ^{1}, θ^{2}, θ^{3}} and the Koszul coaction is

\begin{matrix}S(𝔰𝔩(2)^{∗})_{[2]} → 𝔰𝔩(2) ⊗ S(𝔰𝔩(2)^{∗})_{[2]} \\ θ^{0} ↦ X_{1} ⊗ θ^{1} + X_{2} ⊗ θ^{2} + X_{3} ⊗ θ^{3} \\ θ^{i} ↦ 0,  i = 1, 2, 3 \end{matrix}  (5.87)

We first determine the right 𝔰𝔩(2) action to find the matrices B_{1}, B_{2}, B_{3}. We have

θ^{i} ⊲ X_{j}(X_{q}) = θ^{i} · X_{j}(X_{q}) = θ^{i}([X_{j}, X_{q}]).  (5.88)

Therefore,

B_{1} = ( \begin{matrix}0 & 0 & 0 & 0 \\ 0 & 0 & 0 & −2 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 \end{matrix} ) , B_{2} = ( \begin{matrix}0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 2 \\ 0 & −1 & 0 & 0 \end{matrix} ) , B_{3} = ( \begin{matrix}0 & 0 & 0 & 0 \\ 0 & 2 & 0 & 0 \\ 0 & 0 & −2 & 0 \\ 0 & 0 & 0 & 0 \end{matrix} )

(5.89)

Let A^{1} = (x_{k}^{i}), A^{2} = (y_{k}^{i}), A^{3} = (z_{k}^{i}) represent the 𝔤-coaction on V. According to the above expression of B_{1}, B_{2}, B_{3}, the stability is

∑_{j} A^{j} · B_{j} = ( \begin{matrix}0 & y_{3}^{0} + 2z_{1}^{0} & x_{3}^{0} − 2z_{2}^{0} & −2x_{1}^{0} + 2y_{2}^{0} \\ 0 & y_{3}^{1} + 2z_{1}^{1} & x_{3}^{1} − 2z_{2}^{1} & −2x_{1}^{1} + 2y_{2}^{1} \\ 0 & y_{3}^{2} + 2z_{1}^{2} & x_{3}^{2} − 2z_{2}^{2} & −2x_{1}^{2} + 2y_{2}^{2} \\ 0 & y_{3}^{3} + 2z_{1}^{3} & x_{3}^{3} − 2z_{2}^{3} & −2x_{1}^{3} + 2y_{2}^{3} \end{matrix} ) = 0.  (5.90)

Cyclic Cohomology of Lie Algebras 511

As before, we make the following observations. First,

[B_{1}, A^{1}] = ( \begin{matrix}0 & 0 & −x_{3}^{0} & 2x_{1}^{0} \\ −2x_{0}^{3} & −2x_{1}^{3} & −2x_{2}^{3} − x_{3}^{1} & −2x_{3}^{3} + 2x_{1}^{1} \\ 0 & 0 & −x_{3}^{2} & 2x_{1}^{2} \\ x_{0}^{2} & x_{1}^{2} & x_{2}^{2} − x_{3}^{3} & x_{3}^{2} − 2x_{1}^{3} \end{matrix} ) = 2A^{3} (5.91)

and next

[B_{2}, A^{1}] = ( \begin{matrix}0 & x_{3}^{0} & 0 & −2x_{2}^{0} \\ 0 & x_{3}^{1} & 0 & −2x_{2}^{1} \\ 2x_{0}^{3} & 2x_{1}^{3} + x_{3}^{2} & 2x_{2}^{3} & 2x_{3}^{3} − 2x_{2}^{2} \\ −x_{0}^{1} & −x_{1}^{1} + x_{3}^{3} & −x_{2}^{1} & −x_{3}^{1} − 2x_{2}^{3} \end{matrix} ) = 0  (5.92)

Finally,

[B_{3}, A^{1}] = ( \begin{matrix}0 & −2x_{1}^{0} & 0 & 0 \\ 0 & 0 & 0 & 0 \\ −2x_{0}^{2} & −4x_{1}^{2} & 0 & −2x_{3}^{2} \\ 0 & −2x_{1}^{3} & 0 & 0 \end{matrix} ) = −2A^{1}.  (5.93)

Hence, together with the stability one gets

A^{1} = ( \begin{matrix}0 & x_{1}^{0} & 0 & 0 \\ 0 & 0 & 0 & 0 \\ x_{0}^{2} & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{matrix} )  (5.94)

and

[B_{1}, A^{1}] = ( \begin{matrix}0 & 0 & 0 & 2x_{1}^{0} \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ x_{0}^{2} & 0 & 0 & 0 \end{matrix} ) = 2A^{3}.  (5.95)

Similarly one computes

[B_{1}, A^{2}] = ( \begin{matrix}0 & 0 & 0 & 2y_{1}^{0} \\ −2y_{0}^{3} & −2y_{1}^{3} & 0 & 2y_{1}^{1} \\ 0 & 0 & 0 & 2y_{1}^{2} \\ y_{0}^{2} & y_{1}^{2} & 0 & 2y_{1}^{3} \end{matrix} ) = 0,  (5.96)

as well as

[B_{2}, A^{2}] = ( \begin{matrix}0 & 0 & 0 & −2y_{2}^{0} \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ −y_{0}^{1} & & 0 & 0 \end{matrix} ) = −2A^{3},  (5.97)

and [B_{3}, A^{2}] = 2A^{2}. We conclude that

A^{1} = ( \begin{matrix}0 & c & 0 & 0 \\ 0 & 0 & 0 & 0 \\ d & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{matrix} ) , A^{2} = ( \begin{matrix}0 & 0 & c & 0 \\ d & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{matrix} ) , A^{3} = ( \begin{matrix}0 & 0 & 0 & c \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ \frac{1}{2}d & 0 & 0 & 0 \end{matrix} ) .

(5.98)

Documenta Mathematica 17 (2012) 483–515

One notes that c = 1, d = 0 recovers the Koszul coaction, but obviously it is not the only choice.

6 Cyclic cohomology of Lie algebras

In this section we show that for V, a SAYD module over a Lie algebra 𝔤, the (periodic) cyclic cohomology of 𝔤 with coefficients in V and the (periodic) cyclic cohomology of the enveloping Hopf algebra U(𝔤) with coefficient in the corresponding SAYD over U(𝔤) are isomorphic.

As a result of Proposition 4.2 and Proposition 5.13, we have the following definition.

Definition 6.1. Let 𝔤 be a Lie algebra and V be a right-left SAYD module over 𝔤. We call the cohomology of the total complex of (C_{•}(𝔤, V), ∂_{CE} + b_{K}) the cyclic cohomology of the Lie algebra 𝔤 with coefficients in the SAYD module V, and denote it by HC^{•}(𝔤, V). Similarly we denote its periodic cyclic cohomology by HP^{•}(𝔤, V).

Our main result in this section is an analogous of Proposition 7 of [4].

Theorem 6.2. Let 𝔤 be a Lie algebra and V be a SAYD module over the Lie algebra 𝔤. Then the periodic cyclic homology of 𝔤 with coefficients in V is the same as the periodic cyclic cohomology of U(𝔤) with coefficients in the corresponding SAYD module V over U(𝔤). In short,

HP^{•}(𝔤, V) ≅ HP^{•}(U(𝔤), V).  (6.1)

Proof. The total coboundary of C(𝔤, V) is ∂_{CE} +∂_{K} while the total coboundary of the complex C(U(𝔤), V) computing the cyclic cohomology of U(𝔤) is B + b.

Next, we compare the E_{1} terms of the spectral sequences of the total complexes corresponding to the filtration on the complexes which is induced by the filtration on V via [11, Lemma 6.2]. To this end, we first show that the coboundaries respect this filtration.

As it is indicated in the proof of [11, Lemma 6.2], each F_{p}V is a submodule of V. Thus, the Lie algebra homology boundary ∂_{CE} respects the filtration. As for ∂_{K}, we notice for v ∈ F_{p}V

∂_{K}(X_{1} ∧ ⋯ ∧ X_{n} ⊗ v) = v_{[−1]} ∧ X_{1} ∧ ⋯ ∧ X_{n} ⊗ v_{[0]}  (6.2)

Since

▼(v) = v_{[\overline{−1}]} ⊗ v_{[\overline{0}]} = 1 ⊗ v + v_{[−1]} ⊗ v_{[0]} + ∑_{k≥2} θ_{k}^{−1}(v_{[−k]} ⋯ v_{[−1]}) ⊗ v_{[0]} (6.3)

we observe that v_{[−1]} ∧X_{1}∧⋯∧X_{n}⊗v_{[0]} ∈ ∧^{n+1}𝔤⊗F_{p−1}V. Since F_{p−1}V ⊆ F_{p}V, we conclude that ∂_{K} respects the filtration.

Since the Hochschild coboundary b : V ⊗ U(𝔤)^{⊗n} → V ⊗ U(𝔤)^{⊗n+1} is the alternating sum of cofaces δ_{i}, it suffices to check each δ_{i} preserve the filtration, which is obvious for all cofaces except possibly the last one. However, for the last coface, we take v ∈ F_{p}V and write

v_{[\overline{−1}]} ⊗ v_{[\overline{0}]} = 1 ⊗ v + v_{⟨−1⟩} ⊗ v_{⟨0⟩},  v_{⟨−1⟩} ⊗ v_{⟨0⟩} ∈ 𝔤 ⊗ F_{p−1}V.  (6.4)

We have

δ_{n}(v ⊗ u^{1} ⊗ ⋯ ⊗ u^{n}) = v_{[\overline{0}]} ⊗ u^{1} ⊗ ⋯ ⊗ u^{n} ⊗ v_{[\overline{−1}]} ∈ F_{p}V ⊗ U(𝔤)^{⊗ n+1}. (6.5)

Hence, we can say that b respects the filtration.

For the cyclic operator, the result again follows from the fact that F_{p} is a 𝔤-module. Indeed, for v ∈ F_{p}V

τ_{n}(v⊗u^{1} ⊗⋯⊗u^{n}) = v_{[\overline{0}]} ·u^{1}_{(1)} ⊗S(u^{1}_{(2)})·(u^{2} ⊗⋯⊗u^{n} ⊗v_{[\overline{−1}]}) ∈ F_{p}V ⊗U(𝔤)^{⊗ n}

(6.6)

Finally we consider the extra degeneracy operator

σ_{−1}(v⊗u^{1}⊗⋯⊗u^{n}) = v·u^{1}_{(1)} ⊗S(u^{1}_{(2)})·(u^{2}⊗⋯⊗u^{n}) ∈ F_{p}V ⊗U(𝔤)^{⊗ n} (6.7)

which preserves the filtration again by using the fact that F_{p} is 𝔤-module and the coaction preserve the filtration. As a result now, we can say that the Connes’ boundary B respects the filtration.

Now, the E_{1}-term of the spectral sequence associated to the filtration (F_{p}V)_{p≥0} computing the periodic cyclic cohomology of the Lie algebra 𝔤 is known to be of the form

E_{1}^{j, i}(𝔤) = H^{i+j}(F_{j+1}C(𝔤, V)/F_{j}C(𝔤, V), [∂_{CE} + ∂_{K}])  (6.8)

where, [∂_{CE} +∂_{K}] is the induced coboundary operator on the quotient complex. By the obvious identification

F_{j+1}C(𝔤, V)/F_{j}C(𝔤, V) ≅ C(𝔤, F_{j+1}V/F_{j}V) = C(𝔤, (V/F_{j}V)^{co𝔤}), (6.9)

we observe that

E_{1}^{j, i}(𝔤) = H^{i+j}(C(𝔤, (V/F_{j}V)^{coU(𝔤)}), [∂_{CE}]),  (6.10)

for ∂_{K}(F_{j+1}C(𝔤, V)) ⊆ F_{j}C(𝔤, V).

Similarly,

E_{1}^{j, i}(U(𝔤)) = H^{i+j}(C(U(𝔤), (V/F_{j}V)^{coU(𝔤)}), [b + B]).  (6.11)

Finally, considering

E_{1}^{j, i}(𝔤) = H^{i+j}(C(𝔤, (V/F_{j}V)^{co𝔤}), [0] + [∂_{CE}])  (6.12)

i.e., as a bicomplex with degree +1 differential is zero, the anti-symmetrization map α : C(𝔤, (V/F_{j}V)^{co𝔤}) → C(U(𝔤), (V/F_{j}V)^{coU(𝔤)}) induces a quasiisomorphism [α] : E_{1}^{j, i}(𝔤) → E_{1}^{j, i}(U(𝔤)), ∀i, j by Proposition 7 in [4].

Remark 6.3. In case the 𝔤-module V has a trivial 𝔤-comodule structure, the coboundary ∂_{K} = 0 and

HP^{•}(𝔤, V) = ⨁_{n=• mod 2} H_{n}(𝔤, V).  (6.13)

In this case, the above theorem becomes [4, Proposition 7].

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Bahram Rangipour Department of Mathematics and Statistics University of New Brunswick Fredericton, NB, Canada bahram@unb.ca Serkan Sütlü Department of Mathematics and Statistics University of New Brunswick Fredericton, NB, Canada ssutlu@unb.ca 516

Documenta Mathematica 17 (2012)