In this note, we give a description of the graded Lie algebra of double derivations of a path algebra as a graded version of the necklace Lie algebra equipped with the Kontsevich bracket. Furthermore, we formally introduce the notion of double Poisson-Lichnerowicz cohomology for double Poisson algebras, and give some elementary properties. We introduce the notion of a linear double Poisson tensor on a quiver and show that it induces the structure of a finite dimensional algebra on the vector spaces V_v generated by the loops in the vertex v. We show that the Hochschild cohomology of the associative algebra can be recovered from the double Poisson cohomology. Then, we use the description of the graded necklace Lie algebra to determine the low-dimensional double Poisson-Lichnerowicz cohomology groups for three types of (linear and non-linear) double Poisson brackets on the free algebra in two variables. This allows us to develop some useful techniques for the computation of the double Poisson-Lichnerowicz cohomology.
Attached to a weight space in an integrable highest weight representation of a simply-laced Kac-Moody algebra $\mathfrak{g}$, there are two natural commutative algebras: the cohomology ring of a quiver variety and the center of a cyclotomic KLR algebra. In this note, we describe a natural geometric map between these algebras in terms of quantum coherent sheaves on quiver varieties. The cohomology ring of an algebraic symplectic variety can be interpreted as the Hochschild cohomology of a quantization of this variety in the sense of Bezrukavnikov and Kaledin. On the other hand, cyclotomic KLR a…
For a truncated quiver algebra over a field of arbitrary characteristic, its Hochschild cohomology is calculated. Moreover, it is shown that its Hochschild cohomology algebra is finite-dimensional if and only if its global dimension is finite if and only if its quiver has no oriented cycles.
A partly (anti-)commutative quiver algebra is a quiver algebra bound by an (anti-)commutativity ideal, that is, a quadratic ideal generated by monomials and (anti-)commutativity relations. We give a combinatorial description of the ideals and the associated generator graphs, from which one can quickly determine if the ideal is admissible or not. We describe the center of a partly (anti-)commutative quiver algebra and state necessary and sufficient conditions for the center to be finitely generated as a K-algebra. As an application, necessary and sufficient conditions for finite generation of t…
We present the Gerstenhaber algebra structure on the Hochschild cohomology of Koszul algebras defined by quivers and relations using the idea of homotopy liftings. E.L. Green, G. Hartman, E.N. Marcos and O. Solberg provided a canonical way of constructing a minimal projective bimodule resolution of a Koszul quiver algebra. The resolution has a comultiplicative structure which we use to define homotopy lifting maps. We first present a short example that demonstrates the theory. We then study the Gerstenhaber algebra structure on Hochschild cohomology of a family of bound quiver algebras, some m…
Let k be a field, q in k. We derive a cup product formula on the Hochschild cohomology ring of a family Lambda_q of quiver algebras. Using this formula, we determine a subalgebra of k[x,y] isomorphic to Hochschild cohomology modulo N, where N is the ideal generated by homogeneous nilpotent elements. We explicitly construct non-nilpotent Hochschild cocycles which cannot be generated by lower homological degree elements, thus disproving the Snashall-Solberg finite generation conjecture.
In this paper, we study moduli spaces of representations of certain quivers with relations. For quivers without relations and other categories of homological dimension one, a lot of information is known about the cohomology of their moduli spaces of objects. On the other hand, categories of higher homological dimension remain more mysterious from this point of view, with few general methods. In this paper, we will see how some of the methods used to study quivers can be extended to work for representations of any (noncommutative) monomial algebra with relations of length two. In particular, we…
The main objective of this paper is to provide a theory for computing the Hochschild cohomology of algebras arising from a linear category with finitely many objects and zero compositions. For this purpose, we consider such a category using an ad hoc quiver $Q$, with an algebra associated to each vertex and a bimodule to each arrow. The computation relies on cohomological functors that we introduce, and on the combinatorics of the quiver. One point extensions are occurrences of this situation, and Happel's long exact sequence is a particular case of the long exact sequence of cohomology that w…
The paper includes a new proof of the fact that quiver Grassmannians associated with rigid representations of Dynkin quivers do not have cohomology in odd degrees. Moreover, it is shown that they do not have torsion in homology. A new proof of the Caldero--Chapoton formula is provided. As a consequence a new proof of the positivity of cluster monomials in the acyclic clusters associated with Dynkin quivers is obtained. The methods used here are based on joint works with Markus Reineke and Evgeny Feigin.
For a Poisson algebra $A$, by studying its universal enveloping algebra $A^{pe}$, we prove a duality theorem between Poisson homology and cohomology of $A$.
A version of the twisted Poincaré duality is proved between the Poisson homology and cohomology of a polynomial Poisson algebra with values in an arbitrary Poisson module. The duality is achieved by twisting the Poisson module structure in a canonical way, which is constructed from the modular derivation. In the case that the Poisson structure is unimodular, the twisted Poincaré duality reduces to the Poincaré duality in the usual sense. The main result generalizes the work of Launois-Richard \cite{LR} for the quadratic Poisson structures and Zhu \cite{Zhu} for the linear Poisson structures.
We compute the Hochschild cohomology of the reduced incidence algebras such as the algebra of formal power series, the algebra of exponential power series, the algebra of Eulerian power series, and the algebra of formal Dirichlet series. We achieve the result by carrying out the computation on the coalgebra ${\rm Cotor}$-groups of their pre-dual coalgebras. Using the same coalgebraic machinery, we further identify the Hochschild cohomology groups of an incidence algebra associated to a quiver with the ${\rm Ext}$-groups of the incidence algebra associated to a suspension of the quiver.
Our aim in this paper is to investigate the first Hochschild cohomology of {\em admissible algebras} which can be seen as a generalization of basic algebras. For this purpose, we study differential operators on an admissible algebra. Firstly, differential operators from a path algebra to its quotient algebra as an admissible algebra are discussed. Based on this discussion, the first cohomology with admissible algebras as coefficient modules is characterized, including their dimension formula. Besides, for planar quivers, the $k$-linear bases of the first cohomology of acyclic complete monomial…
We define Calabi-Yau and periodic Frobenius algebras over arbitrary base commutative rings. We define a Hochschild analogue of Tate cohomology, and show that the "stable Hochschild cohomology" of periodic CY Frobenius algebras has a Batalin-Vilkovisky and Frobenius algebra structure. Such algebras include (centrally extended) preprojective algebras of (generalized) Dynkin quivers, and group algebras of classical periodic groups. We use this theory to compute (for the first time) the Hochschild cohomology of many algebras related to quivers, and to simplify the description of known results. Fur…
In this paper, we study the Poisson (co)homology of a Frobenius Poisson algebra. More precisely, we show that there exists a duality between the Poisson homology and the Poisson cohomology, similar to the duality between the Hochschild homology and the Hochschild cohomology of a Frobenius algebra. Using the non-degenerated bilinear form on a Frobenius algebra we construct a Batalin-Vilkovisky structure on the Poisson cohomology ring of a class of Frobenius Poisson algebras.
In order to study the Hochschild cohomology of triangular algebras $\mathcal T$, we construct a spectral sequence, whose terms are parametrized by the length of the trajectories of the quiver associated with $\mathcal T$, and which converges to $HH^*(\mathcal T)$. We explicit its components, and its differentials which are sums of cup products. In case $n=3$, we study some properties of the differential at level 2. Finally, we apply these results to the paths algebra of a quiver without oriented cycles, and link them with previous results on the incidence algebra of a simplicial complex, and m…
Similar to the modular vector fields in Poisson geometry, modular derivations are defined for smooth Poisson algebras with trivial canonical bundle. By twisting Poisson module with the modular derivation, the Poisson cochain complex with values in any Poisson module is proved to be isomorphic to the Poisson chain complex with values in the corresponding twisted Poisson module. Then a version of twisted Poincaré duality is proved between the Poisson homologies and cohomologies. Furthermore, a notion of pseudo-unimodular Poisson structure is defined. It is proved that the Poisson cohomology as a…
We study the noncommutative Poincaré duality between the Poisson homology and cohomology of unimodular Poisson algebras, and show that Kontsevich's deformation quantization as well as Koszul duality preserve the corresponding Poincaré duality. As a corollary, the Batalin-Vilkovisky algebra structures that naturally arise in these cases are all isomorphic.
Double (quasi-)Poisson algebras were introduced by Van den Bergh as non-commutative analogues of algebras endowed with a (quasi-)Poisson bracket. In this work, we provide a study of morphisms of double (quasi-)Poisson algebras, which we relate to the $H_0$-Poisson structures of Crawley-Boevey. We prove in particular that the double (quasi-)Poisson algebra structure defined by Van den Bergh for an arbitrary quiver only depends upon the quiver seen as an undirected graph, up to isomorphism. We derive from our results a representation theoretic description of action-angle duality for several clas…
We study three different (co)homology theories for a family of pullbacks of algebras that we call oriented. We obtain a Mayer Vietoris long exact sequence of Hochschild and cyclic homology and cohomology groups for these algebras. We give examples showing that our sequence for Hochschild cohomology groups is different from the known ones. In case the algebras are given by quiver and relations, and that the simplicial homology and cohomology groups are defined, we obtain a similar result in a slightly wider context. Finally we also study the fundamental groups of the bound quivers involved in t…
We define $H$-Galois extensions for $k$-linear categories and a Hopf algebra $H$ and prove the existence of a Grothendieck spectral sequence for Hochschild-Mitchell cohomology, related to this situation. This spectral sequence is multiplicative and for a $H$ a group algebra decomposes as a direct sum indexed by the set of conjugacy classes of the group. We also compute some Hochschild-Mitchell cohomology groups of categories with infinite associated quiver.
We give a brief review of the cohomological Hall algebra CoHA $\mathcal{H}$ and the K-theoretical Hall algebra KHA $\mathcal{R}$ associated to quivers. In the case of symmetric quivers, we show that there exists a homomorphism of algebras (obtained from a Chern character map) $\mathcal{R}\to \hat{\mathcal{H}}^σ$ where $\hat{\mathcal{H}}^σ$ is a Zhang twist of the completion of $\mathcal{H}$. Moreover, we establish the equivalence of categories of ``locally finite'' graded modules $\hat{\mathcal{H}}^σ-{\rm Mod}_{lf}\simeq \mathcal{R}_{\mathbb Q}-{\rm Mod}_{lf}$. Examples of locally finite $\hat…
We study the Lie algebra structure of the first Hochschild cohomology group of a finite dimensional monomial algebra A, in terms of the combinatorics of its quiver, in any characteristic. This allows us also to examine the identity component of the algebraic group of outer automorphisms of A in characteristic zero. Criteria for the (semi-)simplicity, the solvability, the reductivity, the commutativity and the nilpotency are given.
