Related papers: $L_\infty$-structure on Barzdell's complex for mon…

Given an inclusion $A\hookrightarrow L$ of Lie algebroids sharing the same base manifold $M$, i.e. a Lie pair, we prove that the space $\Gamma(\Lambda^\bullet A^\vee)\otimes_{R} \frac{U(L)}{U(L)\cdot\Gamma(A)}$, where $R=C^\infty(M)$,…

We show that the restricted Lie algebra structure on Hochschild cohomology is invariant under stable equivalences of Morita type between self-injective algebras. Thereby we obtain a number of positive characteristic stable invariants, such…

In this paper, we study compatible Leibniz algebras. We characterize compatible Leibniz algebras in terms of Maurer-Cartan elements of a suitable differential graded Lie algebra. We define a cohomology theory of compatible Leibniz algebras…

We show that the graded commutative ring structure of the Hochschild cohomology HH*(A) is trivial in case A is a triangular quadratic string algebra. Moreover, in case A isgentle, the Lie algebra structure on HH*(A) is also trivial.

Let A be a basic connected finite dimensional algebra over an algebraically closed field k and with ordinary quiver Q without oriented cycle. To any presentation of A by quiver and admissible relations, Martinez-Villa and de La Pena have…

It is well-known that the Kontsevich formality [K97] for Hochschild cochains of the polynomial algebra $A=S(V^*)$ fails if the vector space $V$ is infinite-dimensional. In the present paper, we study the corresponding obstructions. We…

It is proved that the associative differential graded algebra of (polynomial) polyvector fields on a vector space (may be infinite- dimensional) is quasi-isomorphic to the corresponding cohomological Hochschild complex of (polynomial)…

In this paper we study a cohomology theory of compatible Leibniz algebra. We construct a graded Lie algebra whose Maurer-Cartan elements characterize the structure of compatible Leibniz algebras. Using this, we study cohomology,…

We prove that the Gerstenhaber bracket on the Hochschild cohomology of the group algebra of a cyclic group over a field of positive characteristic is not trivial. In this case, we relate the Lie algebra structure on the odd degrees of the…

Let $k$ be a field of odd prime characteristic $p$. We calculate the Lie algebra structure of the first Hochschild cohomology of a class of quantum complete intersections over $k$. As a consequence, we prove that if $B$ is a defect…

Let $G$ be a higher-rank semisimple Lie group over a nonarchimedean local field, for example $G={\rm PGL}(n,Q_P)$. To any lattice $L$ in $G$ there is an associated simplicial complex $B_L$, given by the quotient by $L$ of the Bruhat-Tits…

Let B be a unital Banach algebra. A projection in B which is equivalent to the identitity may give rise to a matrix-like structure on any two-sided ideal A in B. In this set-up we prove a theorem to the effect that the bounded Hochschild…

Recently, relative Rota-Baxter (Lie/associative) algebras are extensively studied in the literature from cohomological points of view. In this paper, we consider relative Rota-Baxter Leibniz algebras (rRB Leibniz algebras) as the object of…

For a finite quiver without sinks, we establish an isomorphism in the homotopy category $\mathrm {Ho}(B_\infty)$ of $B_{\infty}$-algebras between the Hochschild cochain complex of the Leavitt path algebra $L$ and the singular Hochschild…

The Maurer-Cartan algebra of a Lagrangian $L$ is the algebra that encodes the deformation of the Floer complex $CF(L,L;\Lambda)$ as an $A_\infty$-algebra. We identify the Maurer-Cartan algebra with the $0$-th cohomology of the Koszul dual…

In this review article, first we give the concrete formulas of representations and cohomologies of associative algebras, Lie algebras, pre-Lie algebras, Leibniz algebras and 3-Lie algebras and some of their strong homotopy analogues. Then…

We define a cup product on the Hochschild cohomology of an associative conformal algebra $A$, and show the cup product is graded commutative. We define a graded Lie bracket with the degree $-1$ on the Hochschild cohomology $\HH^{\ast}(A)$…

Let ${\rm N}_m(R) = \{ (a_{ij}) \in {\rm M}_m(R) \mid a_{11} = a_{22} = \cdots = a_{mm} \mbox{ and } a_{ij} = 0 \mbox{ for any } i > j \}$ for a commutative ring $R$. Then ${\rm N}_m(R)$ is a quadratic monomial algebra over $R$. We…

We give a general method for constructing explicit and natural operations on the Hochschild complex of algebras over any PROP with $A_\infty$--multiplication---we think of such algebras as $A_\infty$--algebras "with extra structure". As…

We prove $L_{\infty}$-formality for the higher cyclic Hochschild complex $\chH$ over free associative algebra or path algebra of a quiver. The $\chH$ complex is introduced as an appropriate tool for the definition of pre-Calabi-Yau…