Lie structure on the Hochschild cohomology of a family of subalgebras of the Weyl algebra

For each nonzero $h\in \mathbb{F}[x]$, where $\mathbb{F}$ is a field, let $\mathsf{A}_h$ be the unital associative algebra generated by elements $x,y$, satisfying the relation $yx-xy = h$. This gives a parametric family of subalgebras of the Weyl algebra $\mathsf{A}_1$, containing many well-known algebras which have previously been studied independently. In this paper, we give a full description the Hochschild cohomology $\mathsf{HH}^\bullet(\mathsf{A}_h)$ over a field of arbitrary characteristic. In case $\mathbb{F}$ has positive characteristic, the center of $\mathsf{A}_h$ is nontrivial and we describe $\mathsf{HH}^\bullet(\mathsf{A}_h)$ as a module over its center. The most interesting results occur when $\mathbb{F}$ has characteristic $0$. In this case, we describe $\mathsf{HH}^\bullet(\mathsf{A}_h)$ as a module over the Lie algebra $\mathsf{HH}^1(\mathsf{A}_h)$ and find that this action is closely related to the intermediate series modules over the Virasoro algebra. We also determine when $\mathsf{HH}^\bullet(\mathsf{A}_h)$ is a semisimple $\mathsf{HH}^1(\mathsf{A})$-module.


Introduction
Given a field F and a nonzero polynomial h(x) ∈ F[x], let A h be the unital associative F-algebra with two generators x andŷ, subject to the relation yx − xŷ = h. The aim of this article is to describe the structure-given by the Gerstenhaber bracket-of the Hochschild cohomology spaces HH • (A h ) as Lie modules over HH 1 (A h ).
The family A h parametrizes many well-known algebras, which we study simultaneously. For h = 1, we retrieve the first Weyl algebra A 1 . Other particular cases have attracted attention, such as A x , which is the universal enveloping algebra of the two-dimensional non-abelian Lie algebra and A x 2 , known as the Jordan plane, which is a Nichols algebra of non diagonal type. More generally, taking h = x n with n ≥ 3 and setting x in degree 1 andŷ in degree n − 1 then, as observed by Stephenson [11], A x n is Artin-Schelter regular of global dimension two, although it does not admit any regrading so that it becomes generated in degree one.
It is well known that the Weyl algebra is the algebra of differential operators over the one dimensional affine space, where x acts by multiplication and y corresponds to the usual derivative ∂ ∂x . Of course, replacing this last action by h · ∂ ∂x for any fixed polynomial h(x) ∈ F[x] also corresponds to a derivation. If h = 0, the derivation would annihilate everywhere, so we will not consider this case.
There is an embedding of A h in A 1 given by x → x,ŷ → yh, as in [2,Lem. 3.1]. We will thus henceforth takeŷ = yh and consider A h as the unital subalgebra of the Weyl algebra A 1 generated by x andŷ = yh, where [y, x] = 1 and [ŷ, x] = h.
The paper is organized as follows. In Section 2 we prove a few technical lemmas about commutators, while in Section 3 we construct the minimal resolution of A h as an A h -bimodule. In particular, this allows us to give an explicit description of HH 2 (A h ) in positive characteristic. The aim of Section 4 is to complete the construction of a contracting homotopy for the minimal resolution, and in Section 5 we recall the method developed by Suárez-Alvarez [12] to compute the brackets [HH 1 (A), HH n (A)] for any associative unital algebra A. This allows us to obtain in Section 6 the main results of this article: the description, in characteristic zero, of the Lie structure of HH • (A h ) as an HH 1 (A h )-Lie module.
Below we summarize, in simplified from, the main results of the paper. In positive characteristic, an explicit description of HH 2 (A h ) is given in Theorem 3.19, although this is a bit involved. On the other hand, in characteristic zero, HH 2 (A h ) can be presented as a space of polynomials.
Theorem B (Corollary 3.11). Assume char(F) = 0. There are isomorphisms ). In particular, HH 2 (A h ) = 0 if and only if h is a separable polynomial; otherwise, HH 2 (A h ) is infinite dimensional.
The Hochschild cohomology HH • (A h ) = n≥0 HH n (A h ) can be made into a Lie module for the Lie algebra HH 1 (A h ) of outer derivations of A h , under the Gerstenhaber bracket. By the Hochschild-Kostant-Rosenberg Theorem, under suitable assumptions, this bracket is the generalization to higher degrees of the Schouten-Nijenhuis bracket. In our setting this is especially interesting in case char(F) = 0 and gcd(h, h ′ ) = 1 as then the description of HH 1 (A h ) is related to the Witt algebra and, as we shall see, the HH 1 (A h )-Lie module structure of HH 2 (A h ) can be described in terms of the representation theory of the Witt algebra.
Theorem C (cf. Theorem 6.19). Assume that char(F) = 0 and gcd(h, h ′ ) = 1. Let m h + 1 be the largest exponent occurring in the decomposition of h in F[x] into irreducible factors. The structure of HH 2 (A h ) as a Lie module, under the Gerstenhaber bracket, for the Lie algebra HH 1 (A h ) is as follows: (a) There is a filtration of length m h by HH 1 (A h )-submodules HH 2 (A h ) = P 0 P 1 · · · P m h −1 P m h = 0.
such that each factor P i /P i+1 is semisimple. It is noteworthy that, in case F is of characteristic 0 and algebraically closed (so that each irreducible factor of h is linear and the corresponding factor algebra of F[x] is isomorphic to F), then from this theorem and previous results obtained in [1] we can recover the number of irreducible factors appearing in h and the corresponding multiplicities. More specifically, let λ(h) denote the partition encoding the multiplicities of the irreducible factors of h. We can conclude that if λ(h) and λ(g) are different partitions, then A h is not derived equivalent to A g .
We now fix some definitions and notation. Given an associative algebra A and elements a, b ∈ A, we use the commutator notation [a, b] = ab − ba. The center of A and the centralizer of an element a ∈ A will be denoted by Z(A) and C A (a), respectively. An element c ∈ A is normal if cA = Ac (an ideal of A). We remark that the set of normal elements of A forms a multiplicative monoid.
Given a two-sided ideal I of A, we will write a ≡ b (mod I) to mean that a − b ∈ I. This yields an equivalence relation on A with the usual stability properties under addition and multiplication. If J is another ideal such that J ⊆ I, then obviously a ≡ b (mod J) implies a ≡ b (mod I). In case I = cA for some normal element c ∈ A, we also use the notation a ≡ b (mod c).
Unadorned ⊗ will always mean ⊗ F . For any set E, 1 E will denote the identity map on E. Given f ∈ F[x], f (k) stands for the k-th derivative of f with respect to x, which we also denote by f ′ and f ′′ in case k = 1, 2, respectively. If f, g ∈ F[x] are not both zero, then we tacitly assume that gcd(f, g) is monic.
An infinite-dimensional Lie algebra which plays an important role in the description of HH 1 (A) is the Witt algebra. A confusion with terminology may arise here, since the term Witt algebra has been used in the literature to mean two different things: the complex Witt algebra is the Lie algebra of derivations of the ring C[z ±1 ], with basis elements w n = z n+1 d dx , for n ∈ Z; while over a field K of characteristic p > 0, the Witt algebra is defined to be the Lie algebra of derivations of the ring K[z]/(z p ), spanned by w n for −1 ≤ n ≤ p − 2. Here we are considering a subalgebra of the first one (defined over the field F): Acknowledgments: We thank Ken Brown for asking us questions motivating the topic of this paper. We would also like to express our gratitude to Quanshui Wu for kindly providing an argument confirming our conjecture on the description of the Nakayama automorphism of A h .

Some technical results on commutators
In this short section, we gather some technical lemmas about commutators in A h . We will need several additional results on centralizers and commutators in A h from [2], which for convenience we combine below.
(a) We have the following formula for computing in A h : the polynomial algebra in the variables x p and h p y p .
Proof. If char(F) = 0, then the claim follows from [x, Proof. For the first part of (a), if suffices to show that [ŷ, This is clear for j = 0, 1, and for 2 ≤ j ≤ p − 2 we have, using (2.2): which finishes the proof of (a).
is also established and (c) follows from (b), by Proposition 2.1.

Minimal free bimodule resolution of A h
For simplicity, throughout the remainder of this paper, we denote A h simply by A, reserving the notation A h for situations in which we want to emphasize h or make particular choices for h, e.g. when referring to the Weyl algebra A 1 .
In this section, we construct a free resolution of A as an A-bimodule or, equivalently, as a left A e -module, where A e = A ⊗ A op is the enveloping algebra of A and A op is the opposite algebra of A.
We will follow the approach in [4]. Let V = Fx⊕Fŷ be the vector subspace of A spanned by x andŷ and let R = Fr be a vector space of dimension 1. Consider the following sequence of right A-module maps: The maps µ, d 0 and d 1 are in fact A-bimodule maps, whereas the maps s −1 , s 0 and s 1 are just right A-module maps. We describe them all below, except for s 1 , which we discuss only in Section 4: • µ is the multiplication map; x kŷj ⊗ŷ ⊗ŷ ℓ−1−j , with the usual convention that an empty summation is null; in particular, It is easy to check that In fact, we already know that (3.1) is exact, and hence a free resolution of A, since its associated graded complex is exact (see [4]), but it will be useful for further computations to have an explicit contracting homotopy. We claim that the right A-module maps s −1 , s 0 and s 1 form the desired contracting homotopy for (3.1), i.e., that the following hold: 3) The first two equalities are easy to prove and are left as an exercise. So as not to stray from the main ideas of this section, we will defer the construction of the map s 1 and the proof of the last two relations in (3.3) until Section 4 (see Theorem 4.8).
Applying the functor Hom A e (−, A) to the resolution associated with (3.1), we get the commutative diagram where d * i is right composition with d i , for i = 0, 1, and the vector space isomorphisms ρ j are defined as usual by: The maps φ 1 and φ 2 are given by: with the convention that F α (1) = 0. Since F zα = zF α , for z ∈ Z(A), the maps ρ i and φ j are actually Z(A)module maps. It follows that, as a Z(A)-module, the Hochschild cohomology of A can be determined from the maps φ i : • HH 0 (A) = Z(A) = kerφ 1 ; • HH 1 (A) = Der F (A)/Inder F (A) ∼ = kerφ 2 /imφ 1 ; • HH 2 (A) ∼ = A/imφ 2 is the space of equivalence classes of infinitesimal deformations of A (see [6]); • HH i (A) = 0 for all i ≥ 3. The degree zero cohomology HH 0 (A) has been computed in [2,Section 5], while the derivations and the Lie algebra structure of HH 1 (A) were determined in [1], both over arbitrary fields.
• If h = 1, then A 1 is the Weyl algebra and it is well known (see [10]) that HH 0 (A 1 ) = F and HH i (A 1 ) = 0 for all i > 0. In this case, A 1 is graded, setting deg(x) = 1 and deg(y) = −1. • If h = x, then A x is the universal enveloping algebra of the twodimensional non-abelian Lie algebra. In this case, HH 0 (A x ) = F = HH 1 (A x ), by [1,Thm. 5.29]. We will see shortly that HH 2 (A x ) = 0. • If h = x 2 , then A x 2 is the Jordan plane. In this case, A x 2 is graded, setting deg(x) = deg(ŷ) = 1. Note that HH 0 (A x 2 ) = F and by [1,Thm. 5.29], as a Lie algebra, HH 1 (A x 2 ) = Fc ⊕ W, where c is central and W is the Witt algebra given in (1.1). We will see that is naturally a simple module for W and that this module can be embedded into a simple module for the Virasoro algebra.
Our main goal in this section will be to determine the image of φ 2 and the quotient Z(A)-module A/imφ 2 . Later we will determine the Lie action of HH 1 (A) on HH 2 (A) given by the Gerstenhaber bracket. Towards that goal, we start out by studying the map F α given in (3.6). It will be convenient to introduce a mild generalization, so that F α can be defined for all α in the Weyl algebra A 1 ⊇ A. With this extension, the range of F α becomes A 1 , but we will still use F α to denote this map. Lemma 3.8. For α ∈ A 1 , let F α : F[x] −→ A 1 be the linear map defined by (3.6). The following hold for all f, g ∈ F[x]: Proof. To show (a), it suffices to consider f = x k and g = x s , with k, s ≥ 0. Then: This proves (a); (b) is clear and we proceed to prove (c). Again, we need only consider α ∈ A and f = x k , as above. We have: In case char(F) = 0, the following result completely describes the image of the map φ 2 .
Proposition 3.9. The following hold: Since, by Lemma 3.8 (c), Now assume char(F) = 0. By Proposition 2.1, we know that [x, A] = [ŷ, A] = hA and thus imφ 2 2 = [x, A] = hA, which implies that hA ⊆ imφ 2 . Hence, we proceed to show that also h ′ A ⊆ imφ 2 . For α ∈ A, we have seen that  Let us now consider the case char(F) = p > 0. Suppose first that h ∈ F[x p ], a central polynomial. This is a particularly interesting case, not only because it includes the Weyl algebra A 1 , but also since A h is Calabi-Yau if and only if h is central. Indeed, more generally, A h is twisted Calabi-Yau with Nakayama automorphism satisfying x → x,ŷ →ŷ + h ′ , a fact which can be derived from [7,Rmk. 3.4,(2.10)].
Although we can retrieve the following result from Theorem 3.19 below, we think this particular case helps set the stage for our general result and offers a more concrete example.
In particular, in case h = 1 we obtain Proof. We continue to use the maps φ 1 2 and φ 2 2 defined in (3.10). For α ∈ A we have The expression for A/imφ 2 then comes from Lemma 2.4 (b) and Proposition 2.1.
We now tackle the general case for 0 = h ∈ F[x], which is a bit more intricate than the particular case studied above. Consider the decomposition So it remains to determine the image of φ 1 2 | I . Let α ∈ I. Without loss of generality, we can assume that α = zf h p−1 y p−1 with z ∈ Z(A) and f ∈ F[x]. Then, using Lemma 2.4 (a), we have From the above and (3.16) we can conclude that imφ 1 Thence, we obtain a description of HH 2 (A) in positive characteristic.
where J and κ are given in (3.14) and (3.17), respectively, and K is the image of κ. Thus: Remark 3.20. Suppose that in Theorem 3.19 we take by Lemma 2.4 (b), in agreement with the statements in Proposition 3.12.
(a) In case h = 1, then A 1 is the Weyl algebra and, as observed in Proposition 3.12, is a rank-two free module over Z(A 1 ). (b) In case h = x, then A x is the universal enveloping algebra of the two-dimensional non-abelian Lie algebra. We have gcd(h, h ′ ) = 1 we easily see that Hence, Theorem 3.19 yields Then A x 2 is the Jordan plane. We distinguish between two cases: In this case x 2 is central and we use Proposition 3.12 to obtain the isomorphism In this case x 2 is not central and we use Theorem 3.19. Since where the last summand is zero in case p = 3. Hence Theorem 3.19 gives Notice that in all cases, HH 2 (A x 2 ) is not a free module over Z(A x 2 ), although it is composed of a torsion summand and a free summand of rank one.
We have seen in the examples that, in general, HH 2 (A) is not a free module over Z(A). The next theorem provides a necessary and sufficient condition for HH 2 (A) to be free. Proof. The last statement follows from the first by [1,Thm. 6.29], so we need only focus on HH 2 (A).
The condition gcd(h, h ′ ) = 1 is necessary, as otherwise J/gcd(h, h ′ )J would be nonzero and annihilated by the central element (gcd(h, h ′ )) p . Next we prove that it is sufficient.
and it follows that f = κ(q) ∈ K.
. But t is a divisor of h and gcd(h, h ′ ) = 1 so it follows that g ∈ tF[x], as required.
Take q, r ∈ F[x] with g = ωq + rh. Applying κ to this equality we obtain κ(g) = ωκ(q) + κ(rh) and thus ω divides κ(rh). So if suffices to prove that if ω divides κ(rh) then rh ∈ ωF[x] + hF[x p ]. In other words, we may assume without loss of generality that g = rh. Write Proof of subclaim 2: Note that κ(rh) = r ′ h 2 , so we need to show that if ω divides r ′ h 2 , then ω divides rh. From this point on, our proof follows that of [1, Lem. 6.28 (iv)], although the details are a bit more intricate and some modifications are needed. Thus, we suspend the proof of the subclaim here and refer the interested reader to the proof of [1, Lem. 6.28 (iv)].
By the above arguments, the claim is also established, thus proving the Theorem. 4. The contracting homotopies s −1 , s 0 and s 1 Recall the definition of the right A-module maps s −1 and s 0 , given at the beginning of Section 3. In this section we prove the two final relations in (3.3), together with a few other useful identities.
Proof. To prove (a), notice that, by the definition of s 0 , we have Since s 0 is a right A-module map, we can take b = 1 and by linearity we can further assume that f = x j and a = x kŷℓ . Then: As above, it suffices to prove (b) for α = a ⊗ v ⊗ 1. Using (a), we have: Recall that we have fixed r as the basis element of the one-dimensional vector space R. Consider the linear map G : is the A-bimodule map which sends both 1 ⊗ x ⊗ 1 and 1 ⊗ŷ ⊗ 1 to 1 ⊗ r ⊗ 1. Thus, by Lemma 4.1 (a), G is a derivation.
We deduce that d 1 • G is also a derivation. Define D : To prove the claimed identiy, it suffices to show that D is also a derivation and that d 1 • G(x) = D(x). The latter is easy to verify, so we turn to proving that D is a derivation, which is also straightforward, using the properties of s 0 : We are finally ready to define the homotopy s 1 : Proof. We start by showing that the claimed equality holds for elements of the formŷ ℓ ⊗ x ⊗ 1, by induction on ℓ ≥ 0. In case ℓ = 0 we have Next, assume that the result holds for elements of the formŷ k ⊗ x ⊗ 1, with k ≤ ℓ. Using (2.2) we have Also, by the inductive definition of s 1 and the fact that d 1 is a bimodule map, By the induction hypothesis we havê Finally, using Lemma 4.3, it follows that The term x ⊗ 1)) can be further expanded as follows: Combining all of these expressions, we see easily that all terms cancel out except for the termŷ ℓ+1 ⊗x⊗1 in the expansion ofŷd 1 (s 1 (ŷ ℓ ⊗x⊗1)) above, so we have the desired identity (s 0 By the equality s 0 (f d 0 (α)) = f s 0 (d 0 (α)), for f ∈ F[x] and α ∈ A ⊗ V ⊗ A, proved in Lemma 4.1, and the definition of s 1 , we conclude that (s 0 and a ∈ A. So next we focus on elements of the form fŷ ℓ ⊗ŷ ⊗ a. We will make use of the identity s 0 (ŷ ℓ+1 ⊗ 1−ŷ ℓ ⊗ŷ) =ŷ ℓ ⊗ŷ ⊗ 1 to perform the required calculation. Then, Combining all of the above, we have proved the claim. Now we aim to prove the last relation in (3.3), namely s 1 • d 1 = 1 A⊗R⊗A . We start with a technical identity which just depends on the fact that G and δ are derivations.
Lemma 4.5. Given k ≥ 1 and r ≥ 0, Now recall that for any derivation D, the generalized Leibniz rule says that . So the right-hand side of the running equality is Since G(δ m (1)) = 0 for all m ≥ 0 and δ r−m (1) = 0 for all m < r, the latter expression is just G(δ r (x k )), as desired.
Our next results concern the computation of s 1 .
Proposition 4.6. For all ℓ ≥ 0 and all f ∈ F[x], the following identity holds: Proof. By linearity, it is enough to show the identity for all k ≥ 0. This holds trivially if k = 0, so we assume that k ≥ 1.
Firstly, let us observe that by the relationŷf = fŷ + δ(f ) and the recurrence relation for s 1 , it follows that for all f ∈ F[x] and ℓ ≥ 0. Thus, using (2.2), we have, for 0 ≤ i ≤ k − 1: Hence, and it remains to prove that Using (2.2), we can write the former as Let a = j + t + m. Notice that 0 ≤ a ≤ ℓ and that the sum above can be written as Therefore, we just need to prove that φ(i, j, t) = ℓ a G(δ a (x k )). Since ℓ j by Lemma 4.5 we deduce that Hence, the result is established.
In particular, taking f = x, we obtain the following explicit formula for s 1 : Proof. If ℓ = 0, Proposition 4.6 yields s 1 (ŷs 0 (f ⊗ 1)) = G(f ), which agrees with the formula we are proving. We proceed inductively, using Proposition 4.6: Finally, we can prove the main result of this section. Proof. It remains to prove the identity s 1 • d 1 = 1 A⊗R⊗A from (3.3), and it clearly suffices to check this identity on elements of the formŷ ℓ ⊗ r ⊗ 1, as s 1 is also a left F[x]-module homomorphism. The case ℓ = 0 is straightforward, so assume that ℓ ≥ 1. Then and by Proposition 4.7, we have Using adequate combinatorial identities, we obtain which proves the desired identity.

The Gerstenhaber bracket: general remarks
The Hochschild cohomology HH • (A) = n≥0 HH n (A) has a rich structure, including an associative, graded-commutative product (relative to homological degree), given by the cup product, and also a graded Lie bracket [ , ] of (homological) degree −1; these are related by the graded Poisson identity. In particular, the graded anti-symmetric property of [ , ] means [α, β] = −(−1) (m−1)(n−1) [β, α] , for all α ∈ HH m (A) and β ∈ HH n (A), and there is a corresponding graded version of the Jacobi identity (see [5]). Under this construction, HH • (A) becomes a Gerstenhaber algebra. In particular, the Jacobi identity implies that HH • (A) is a Lie module for the Lie algebra HH 1 (A), extending the usual Lie bracket of derivations on HH 1 (A). In case A is a smooth finitely-generated F-algebra and F is perfect, the Hochschild-Kostant-Rosenberg Theorem gives an isomorphism of Gerstenhaber algebras, telling that, in this situation, the Gerstenhaber bracket is the generalization to higher degrees of the Schouten-Nijenhuis bracket.
The Gerstenhaber structure of Hochschild cohomology is particularly interesting for us since in case char(F) = 0 and gcd(h, h ′ ) = 1, the description of HH 1 (A) involves the Witt algebra W. In prime characteristic, most of the computations of the Gerstenhaber structure in Hochschild cohomology concern group algebras and tame blocks, see for example [3,9].
Although the Gerstenhaber bracket does not depend on the chosen bimodule projective resolution of A, it is in general difficult to compute it on an arbitrary resolution other than the bar resolution. In spite of this, we always have D, z = D(z) and D, D ′ = [D, D ′ ] for D, D ′ ∈ Der F (A) and z ∈ Z(A), so it remains to compute HH 1 (A), HH 2 (A) , which is what we undertake in this section. Notice that, in our case, we already have the contracting homotopy of the minimal resolution, from which the comparison maps can be obtained. Nevertheless, we will use an easier method that, for the family of algebras we consider, also needs the contracting homotopy.
To avoid cumbersome notation, we identify D ∈ Der F (A) with its canonical image D ∈ HH 1 (A). We will often refer to the map [D, −] : HH i (A) −→ HH i (A) as the (Lie) action of D ∈ HH 1 (A) on HH i (A).

5.1.
The method of Suárez-Álvarez for computing HH 1 (A), − . In this subsection, we will describe a method devised by Suárez-Álvarez in [12] to compute the Gerstenhaber bracket HH 1 (A), − in terms of an arbitrary projective resolution of A as a bimodule. The reader is advised to consult [12] for further details and all the proofs.
of M , a ψ-lifting of the ψ-operator f to P • is a sequence f • = (f i ) i≥0 of ψ-operators f i : P i −→ P i such that the following diagram commutes: It was shown in [12, Lem. 1.4] that every ψ-operator f admits a unique (up to B-module homotopy) ψ-lifting. Given a ψ-operator f and a ψ- Going back to the case under study, with A = A h , ǫ = µ (the multiplication map), P 0 = A ⊗ A, P 1 = A ⊗ V ⊗ A and P 2 = A ⊗ R ⊗ A, it can be checked that D • µ = µ • D e and D e is trivially a D e -operator on A ⊗ A, so we can choose D 0 = D e . Taking i = 2 and using the map ρ 2 from Section 3 to identify HH 2 (A) with a homomorphic image of A, we obtain the formula describing the Lie action of HH 1 (A) on HH 2 (A): for a ∈ A and D ∈ Der F (A), where χ a ∈ Hom A e (A ⊗ R ⊗ A, A) is defined by χ a (1 ⊗ r ⊗ 1) = a.

5.2.
The D e -lifting of D to (3.1). In order to make use of (5.1), it remains to determine the D e -lifting D 2 of D, which we do in this subsection. We begin with a few general observations aimed at simplifying computations, then we determine the D e -liftings D 1 and D 2 . The proof of the lemma that follows is standard and is thus omitted.  Throughout the rest of this subsection, fix D ∈ Der F (A) and let D 0 = D e : A e −→ A e . Next we define a D 0 -lifting D 1 : 1⊗ŷ⊗1} is a free basis for A⊗V⊗A as an A e -module, Lemma 5.2(a) guarantees the existence of a unique D 0 -operator, which we still denote by D 1 , defined on A ⊗ V ⊗ A and extending the above rule.
First, notice that by linearity of D and s 0 , one has As s 0 is a right A-module map, this expression matches the one in the statement. Now, by Lemma 5.2 (b), it suffices to check the equality D 0 • d 0 = d 0 • D 1 on elements of the form 1 ⊗ v ⊗ 1. Thus, using the second identity in (3.3), we establish the final claim: Before we proceed to define the D 0 -lifting D 2 , we prove some auxiliary relations which will simplify several expressions, including one for D 2 (1 ⊗ r ⊗ 1).
, where this sum is understood to be 0 in case k ∈ {0, 1}.
Finally, for the proof of (d) we have, using the definition of D 1 , parts (a) and (b) and the definition of s 1 : Motivated by Lemma 5.4(c), we extend the map G linearly to A, by setting We are now ready to define the D 0 -operator D 2 in terms of D 1 and the homotopy s 1 .

Proof. By Lemma 5.2 (a), there exists a unique
can be computed as in the proof of Lemma 5.3. Now, using Lemma 5.4 and (5.6), we have Finally, by Lemma 5.2 (b), it is enough to show that ⊗ 1), so we compute, using Lemma 4.4 and Lemma 5.3:

Technical lemmas.
We need to prove yet some more technical results which will allow us to simplify the computation of the Gerstenhaber bracket given in (5.1). Although these will be particularly useful in case char(F) = 0, most statements hold over an arbitrary field, so we include them here. Following [1, Lem. 2.13], it will be useful to define, for 0 = f ∈ F[x], the element π f such that: , up to a nonzero scalar. In this subsection we will mostly work over some homomorphic image of A and we will extensively use the notations a ≡ b (mod I) and a ≡ b (mod c), defined in the introduction to mean that a − b ∈ I and a − b ∈ cA = Ac, for a two-sided ideal I and a normal element c, respectively. We remark that the monoid of normal elements of A was described in [2,Thm. 7.2] and, in particular, any product of factors of h is normal in A. Lemma 5.9. Let D ∈ Der F (A), a ∈ A and k ≥ 0. The following hold: (a) D(h) ∈ hA and D(x) ∈ π h A; Proof. The defining relation for A implies that Since for any f ∈ F[x] we have h divides h ′ f if and only if π h divides f , we conclude that D(x) ∈ π h A, finishing the proof of (a).
Let g = gcd(h, h ′ ). Up to a nonzero scalar, h = π h g. Write D(x) = π h b for some b ∈ A. By (b), As h ′ = π h g ′ + π ′ h g and g divides h ′ , we deduce that g divides π h g ′ , so D(g) ∈ gA + hA = gA.
establishing the first claim in (c). Thus, for all ℓ ≥ 0, . By Lemma 5.4 we have: By (a) and (b), k−1 i=1 χ•s 1 (θ i ⊗x⊗x k−i−1 ) ∈ hA. Thus, working modulo hA and using the commutativity of A/hA and the hypothesis that char(F) = 2, we obtain In particular, Lastly, we prove (e) by induction on ℓ ≥ 0. As χ • s 1 (1 ⊗ x ⊗ 1) = 0, the base step is established and we assume that holds for some ℓ ≥ 0. Then, by the definition of s 1 , the commutativity of A/gcd(h, h ′ )A and part (c) above, as δ j (x) ∈ hA for all positive j, and k ≥ 0. Then: . (Notice that in case k = 0 the above expression still makes sense, as Proof. Working modulo hA, we deduce (a): In particular, multiplying both sides of (a) by gcd(h, h ′ ) = h/π h we obtain and it follows that h k y k+1 , h ∈ gcd(h, h ′ )A.
We are now ready to prove (b) by induction on k ≥ 0, the base case being trivial. Supposing that (b) holds for a certain k ≥ 0, we get We also prove (c) by induction on k ≥ 0. The case k = 0 is immediate from Lemma 5.10(c). For the inductive step, assume the congruence holds for k ≥ 0. By (5.12) we have By Lemma 5.10 (c),

The Gerstenhaber bracket
In this section we determine the structure of HH 2 (A) as a module over the Lie algebra HH 1 (A) under the Gerstenhaber bracket, always under the assumption that char(F) = 0. We will prove some of the main results of this article. In the first subsection we will describe two different subspaces of the space of linear derivations of our algebra, that will act on HH 2 (A) in a very different way. Next we will describe the action of the classes of these derivations on HH 2 (A). Then we achieve our goal of giving an explicit description of HH 2 (A) as HH 1 (A)-Lie module. We finish the section by relating this action of HH 1 (A) on HH 2 (A) with the representation theory of the Virasoro algebra, and then by discussing several special cases.
6.1. The Lie algebra structure of HH 1 (A). The Lie algebra structure of HH 1 (A) in case char(F) = 0 is described explicitly in [1, Sec. 5] and we briefly collect the results we need below.
There are two types of derivations of A, which together describe Der F (A) and HH 1 (A): • For any g ∈ F[x], let D g be the derivation of A such that D g (x) = 0 and D g (ŷ) = g. Then, {D g | g ∈ F[x]} is an abelian Lie subalgebra of Der F (A) and D g ∈ Inder F (A) if and only if g ∈ hF[x]. • Viewing, as usual, A = A h ⊆ A 1 withŷ = yh, define the elements a n = π h h n−1 y n ∈ {u ∈ A 1 | [u, A] ⊆ A} (the normalizer of A in A 1 ), for all n ≥ 1. It will also be convenient to consider the element a 0 = π h /h = By [1,Lem. 4.14], ad ga 0 = −D δ 0 (g) . For notational simplicity, by [2, Thm. 8.2], we can assume that h is monic, say h = u α 1 1 · · · u αt t , where u 1 , . . . , u t are the distinct monic prime factors of h, with multiplicities α 1 , . . . , α t . Up to changing the order of the factors, we can further assume that there is 0 ≤ k ≤ t such that α 1 , . . . , α k ≥ 2 and α k+1 = · · · = α t = 1. Moreover, if k = 0 then gcd(h, h ′ ) = 1 and in this case HH 2 (A) = 0, so there is nothing to prove.
is a field extension of the Witt algebra. ). We will use (5.1) and also the identification introduced there between A/gcd(h, h ′ )A and Hom A e (A ⊗ R ⊗ A, A)/im d * 1 , which associates the element a ∈ A with the map χ a ∈ Hom A e (A ⊗ R ⊗ A, A) defined by χ a (1 ⊗ r ⊗ 1) = a, and similarly for the corresponding homomorphic images. Now, Lemma 5.10(d) implies that for all a ∈ A, the image of χ a • s 1 • D 1 • s 0 (h ⊗ 1) in HH 2 (A) is zero. Thus we have, using Lemma 5.7, for all a ∈ A and D ∈ Der F (A). Moreover, by Lemma 5.9(c), the image of D(a) in HH 2 (A) depends only on the class a + gcd(h, h ′ )A and similarly, χ a (G(D(x))) and χ a (s 1 (D(ŷ) ⊗ x ⊗ 1)) depend only on the classes D(x) + hA and D(ŷ) + gcd(h, h ′ )A, respectively, by Lemma 5.10. We will first consider the derivations of the form D g , for g ∈ F[x]. Fix g and let D = D g . Take a = p(x)ŷ k for some p(x) ∈ F[x] and k ≥ 0. Then D(x) = 0 = s 1 (D(ŷ) ⊗ x ⊗ 1) and by Lemma 5.9,D(p(x) In particular, Z(HH 1 (A)), HH 2 (A) = 0, by Theorem 6.2(b). Now we can turn our attention to the derivations of the form ad gan , with g ∈ F[x] and n ≥ 0. Proof. We have D(x) = π h gh n−1 y n , x = nπ h gh n−1 y n−1 ≡ nπ h gŷ n−1 (mod gcd(h, h ′ )), where the last congruence comes from Lemma 5.11(b). Also, D(ŷ) = π h gh n−1 y n ,ŷ = π h gh n−1 y n+1 h − yπ h gh n y n = π h gh n y n+1 + π h gh n−1 y n+1 , h − π h gh n y n+1 − [y, π h gh n ] y n ≡ (n + 1)π h h ′ gh n−1 y n + n + 1 2 π h gh ′′ h n−1 y n−1 − (π h gh n ) ′ y n (mod h) ≡ −δ 0 (g)ŷ n (mod gcd(h, h ′ )), using Lemma 5.11(a) and (b), the fact that gcd(h, h ′ ) divides h ′′ π h and (6.1).
Finally, using Lemma 5.9 (b), Hence, for D = ad gan and a = p(x)ŷ k ∈ A, we can now compute [D, a] as an element of D[ŷ], using (6.3), Lemma 5.10(e), Lemma 5.11(c) and recalling that gcd(h, h ′ ) divides h ′′ π h : It thus follows that, working in HH 2 (A) = A/gcd(h, h ′ )A and recalling (6.1): Therefore, we have proved the main result of this subsection.
Theorem 6.6. Assume that char(F) = 0. The Lie action of HH 1 (A) on HH 2 (A) under the Gerstenhaber bracket is given by the following formulas: for all g ∈ F[x] and n ≥ 0, where a n = π h h n−1 y n .
We make the identification and there exist nonzero pairwise orthogonal idempotents e 1 , . . . , e k ∈ D with e 1 + · · · + e k = 1, D = De 1 ⊕ · · · ⊕ De k and (these isomorphisms are both as algebras and as left F[x]-modules). Define . Then, by Theorem 6.2(d), we have As the notation suggests, the algebra D is a quotient of D by the ideal u 1 · · · u k D. Let e 1 , . . . , e k ∈ D be the images of the idempotents e 1 , . . . , e k ∈ D under the canonical epimorphism. It is straightforward to see that these are still nonzero pairwise orthogonal idempotents in D with e 1 + · · · + e k = 1, D = De 1 ⊕ · · · ⊕ De k and Thus, Θ 0 = 1, Θ 1 = u 1 · · · u k = π (h/π h ) and for any i ≥ m h , Θ i = gcd(h, h ′ ). Finally, define P i = Θ i D[ŷ] ⊆ HH 2 (A). We record a few useful facts below. Lemma 6.10. For i ≥ 0,we have: is a Lie HH 1 (A)-submodule of HH 2 (A) and there is a strictly decreasing filtration Proof. (a) is clear from the definition. The identity in (b) holds trivially for i = 0 and we prove it by induction on i ≥ 0. So assume that , by (a), we have The fact that (6.11) is a decreasing filtration of vector spaces is clear because Θ i divides Θ i+1 . Since the quotient α j ≥i+2 u j of these polynomials is not a unit, for 0 ≤ i ≤ m h − 1, by the definition of m h , the filtration is strict. Thus, it remains to show that [ad gan , P i ] ⊆ P i , for all g ∈ F[x] and n, i ≥ 0. By (6.8), given f ∈ F[x] and ℓ ≥ 0: Set S i = P i /P i+1 , for 0 ≤ i ≤ m h − 1. We have seen that S i is a nonzero HH 1 (A)-module under the action induced from the Gerstenhaber bracket. Noting that δ 0 (g) = gδ 0 (1) + g ′ π h (see [1,Lem. 4.14]) and π h Θ i ∈ Θ i+1 F[x], we see that this action is completely described by the following computation in S i : By the above isomorphisms, the element Our next step is to describe the Lie algebra isomorphism (6.9). We will need the following. Lemma 6.14. There is an element ν ∈ F[x], determining a unique class modulo ). For such an element, the following hold: and gcd(δ 0 (1), Θ 1 ) = 1. This shows the existence of ν with νδ 0 (1) ≡ 1 (mod Θ 1 F[x]) and also proves (a).
Based on the proof of [1,Lem. 5.19] and the definition of D q , we can deduce that under the isomorphism (6.9), the element ge q ⊗ w m ∈ D q ⊗ W is mapped to −ad geqνa m+1 + N ∈ [HH 1 (A), HH 1 (A)]/N, for 1 ≤ q ≤ k, g ∈ F[x] and m ≥ −1, where ν is as in Lemma 6.14. Using these identifications and those in (6.13), we have: if α q ≤ i + 1, by (6.12) and Lemma 6.14, as Θ i+1 divides Θ 1 Θ i . Moreover, we can use Lemma 6.14(b) since u q e q = 0 in D q , yielding: The above shows that D q ⊗ W acts trivially on D j [ŷ] ⊆ S i except if j = q and α q ≥ i + 2. In the latter case, the action of D q ⊗ W on D q [ŷ] is given by In particular, each D j [ŷ] ⊆ S i in the decomposition (6.13) is an HH 1 (A)submodule of S i . Notice that in (6.15), the elements f e q and ge q are scalars in the field extension D q ∼ = F[x]/u q F[x] of F and the action (6.15) is D q -linear. This motivates the following definition. Fix a scalar µ ∈ F and let V µ = F[ŷ]. Define an action of the Witt algebra W on V µ by (6.16) w m .ŷ ℓ = (ℓ − (m + 1)µ)ŷ m+ℓ , for all m ≥ −1 and ℓ ≥ 0.
It can be verified that this indeed defines an action of W on V µ , for any µ ∈ F (for µ of the form α−i α−1 with α ≥ i + 2 this statement is implied by (6.15)).
The module V µ is related to the intermediate series modules for the Witt and Virasoro algebras (compare (6.21), ahead). Next, we record irreducibility and isomorphism criteria for these modules. Lemma 6.17. For F an arbitrary field of characteristic 0 and µ ∈ F, let V µ be the W-module defined in (6.16). Then: Proof. The proof is straightforward, so we just sketch it. First, if µ = 0 then Fŷ 0 is a submodule of V 0 , so V 0 is reducible. Suppose now that µ = 0. Let X be a nonzero submodule of V µ . Since w ℓ −1 .ŷ ℓ = ℓ!ŷ 0 , it follows by the usual argument thatŷ 0 ∈ X. Taking into account that w m .ŷ 0 = −(m+1)µŷ m ∈ X for all m ≥ 0 and µ = 0, we deduce that X = V µ . Thus V µ is irreducible and (a) is proved.
The action of w 0 on V µ is diagonalizable with eigenvalues {ℓ − µ} ℓ≥0 , with −µ being the unique eigenvalue such that −µ − 1 is no longer an eigenvalue. Thus the action of W on V µ determines µ, which proves (b).
It follows from the above that for all 0 ≤ i ≤ m h − 1 and all j such that α j ≥ i + 2, the D j ⊗ W-module D j [ŷ] ⊆ S i is irreducible and it is isomorphic to D j ⊗ V µ ij , where µ ij = α j −i α j −1 = 0. As the action depends on i, it is convenient to introduce i into the notation for this module. Thus, we henceforth denote this module by V ij : for all 0 ≤ i ≤ m h − 1 and j such that α j ≥ i + 2. Moreover, D q ⊗ W acts trivially on V ij for q = j, so it follows by Theorem 6.2 and (6.9) that V ij is an irreducible HH 1 (A)-submodule of S i on which both Z(HH 1 (A)) and the nilpotent radical N of [HH 1 (A), HH 1 (A)] act trivially. As a result of this analysis, we conclude that S i is a completely reducible HH 1 (A)-module with semisimple decomposition (cf. (6.13)): We summarize these results in the following, which constitutes the main result of this paper. Let h = u α 1 1 · · · u αt t be the decomposition of h into irreducible factors with 0 ≤ k ≤ t such that α 1 , . . . , α k ≥ 2 and α k+1 = · · · = α t = 1. Since HH 2 (A) = 0 if and only if k ≥ 1, we assume that k ≥ 1 and set m h = max{α j − 1 | 1 ≤ j ≤ k}.
The structure of HH 2 (A) as Lie module over the Lie algebra HH 1 (A) under the Gerstenhaber bracket is as follows: (a) There is a filtration of length m h by HH 1 (A)-submodules HH 2 (A) = P 0 P 1 · · · P m h −1 P m h = 0.
(b) For each 0 ≤ i ≤ m h −1 the factor module S i = P i /P i+1 is completely reducible with semisimple decomposition S i = α j ≥i+2 V ij , where: (i) The nilpotent radical Z(HH 1 (A))⊕ N of HH 1 (A) acts trivially on S i , so S i becomes a D 1 ⊗ W ⊕ · · · ⊕ D k ⊗ W -module, where Proof. All of the above statements have been proved, except for (iv) and (d). We start with (iv). If V ij ∼ = V i ′ j ′ then D j ⊗ W acts non-trivially on V i ′ j ′ , so j = j ′ , by (iii). Thus, by Lemma 6.17(b), µ ij = µ i ′ j , which in turn implies i = i ′ . For the proof of (d), if h is not divisible by the cube of any non-constant polynomial then m h = 1 and HH 2 (A) = S 0 , which we have seen in (b) is semisimple. Conversely, if m h ≥ 2 then there is some i such that α i ≥ 3, say i = 1. By (6.8), because u 2 1 divides gcd(h, h ′ ) but it does not divide ad u 1 ···u k a 1 ,ŷ 0 . But ad u 1 ···u k a 1 ∈ N and N annihilates all the composition factors of HH 2 (A), by (i), so HH 2 (A) cannot be semisimple in this case.
Before we proceed to illustrate our result with some special cases, we first want to establish a connection between the representations V ij and the Virasoro algebra. Recall that the Virasoro algebra is the unique (up to isomorphism) central extension of the full Witt algebra of derivations of F[z ±1 ]. This Lie algebra is defined as Vir = i∈Z F.w i ⊕ F.c, where [c, Vir] = 0 and [w m , w n ] = (n − m)w m+n + δ m+n,0 m 3 − m 12 c ∀m, n ∈ Z.
The module U µ is an intermediate series module (see [8] for details).
The following can be readily checked by the reader: We continue to assume that char(F) = 0.
Example 6.22 (h = x n ). Let's consider the case where h has a unique irreducible factor. For the sake of simplicity, we will assume that this factor is x, that is, h = x n with n ≥ 2; the more general case of an irreducible factor of higher degree is entirely analogous. In this case: Z(HH 1 (A x n )) = FD x n−1 , where D x n−1 (x) = 0 and D x n−1 (ŷ) = x n−1 , [HH 1 (A x n ), HH 1 (A x n )]/N ∼ = W (the Witt algebra), For 0 ≤ i ≤ n − 1, let P i = x i D[ŷ], so that we get the following filtration of HH 1 (A x n )-submodules of HH 2 (A x n ) HH 2 (A x n ) = P 0 P 1 · · · P n−2 P n−1 = 0.
Thus, S i ∼ = V n−i n−1 is simple and the composition factors {S i } 0≤i≤n−2 of HH 2 (A x n ) are pairwise non isomorphic. In particular, HH 2 (A x n ) has length n − 1 as a HH 1 (A x n )-module, with distinct composition factors.
The next example, a particular case of the previous one, focuses on the Jordan plane.
Example 6.23 (The Jordan plane). Taking h = x 2 , we obtain the algebra A x 2 , known as the Jordan plane, with homogeneous defining relationŷx = xŷ + x 2 . The description here is: where D x (x) = 0, D x (ŷ) = x and W is the Witt algebra.
It follows that HH 2 (A x 2 ) is a simple HH 1 (A x 2 )-module annihilated by D x and such that, as a W-module, HH 2 (A x 2 ) ∼ = V 2 .
Our last example deals with the case where HH 2 (A) is a semisimple Lie module.
Example 6.24 (h is cube free). By Theorems 6.2 and 6.19 (d), the following conditions are equivalent: • HH 2 (A) is a semisimple HH 1 (A)-module; • N = 0; • HH 1 (A) is a reductive Lie algebra; • h is cube free. Here we study the case in which these conditions hold, so the decomposition of h into irreducible factors is of the form h = u 2 1 · · · u 2 k u k+1 · · · u t , for some 1 ≤ k ≤ t. We have dim F Z(HH 1 (A)) = deg u 1 · · · u t , HH 1 (A) = Z(HH 1 (A)) ⊕ (D 1 ⊗ W) ⊕ · · · ⊕ (D k ⊗ W), and W is the Witt algebra.
Then, Z(HH 1 (A)) acts trivially on HH 2 (A) and D i ⊗ W acts trivially on D j [ŷ], if i = j. As a D j ⊗ W-module, D j [ŷ] ∼ = D j ⊗ V 2 . Thus the irreducible summands of HH 2 (A) are D j [ŷ] 1≤j≤k , they are pairwise non-isomorphic as HH 1 (A)-modules and the composition length of HH 2 (A) is k.