Related papers: Infinity Algebras, Massey Products, and Deformatio…

A relative Rota-Baxter algebra is a triple $(A, M, T)$ consisting of an algebra $A$, an $A$-bimodule $M$, and a relative Rota-Baxter operator $T$. Using Voronov's derived bracket and a recent work of Lazarev et al., we construct an…

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…

A universality of deformed Heisenberg algebra involving the reflection operator is revealed. It is shown that in addition to the well-known infinite-dimensional representations related to parabosons, the algebra has also finite-dimensional…

Given an associative multiplication in matrix algebra compatible with the usual one or, in other words, linear deformation of matrix algebra, we construct a solution to the classical Yang-Baxter equation. We also develop a theory of such…

This article explores \Z_2-graded L_\infinity algebra structures on a 2|1-dimensional vector space. The reader should note that our convention on the parities is the opposite of the usual one, because we define our structures on the…

Let $A$ be a star product on a symplectic manifold $(M,\omega_0)$, $\frac{1}{t}[\omega]$ its Fedosov class, where $\omega$ is a deformation of $\omega_0$. We prove that for a complex polarization of $\omega$ there exists a commutative…

We study possible decompositions of totally decomposable algebras with involution, that is, tensor products of quaternion algebras with involution. In particular, we are interested in decompositions in which one or several factors are the…

In this article we develop an approach to deformations of the Witt and Virasoro algebras based on $\sigma$-derivations. We show that $\sigma$-twisted Jacobi type identity holds for generators of such deformations. For the $\sigma$-twisted…

By means of contractions of Lie algebras, we obtain new classes of indecomposable quasi-classical Lie algebras that satisfy the Yang-Baxter equations in its reformulation in terms of triple products. These algebras are shown to arise…

Let $\Lambda$ be a graded self-injective algebra. We describe its smash product $\Lambda# k\mathbb Z^*$ with the group $\mathbb Z$, its Beilinson algebra and their relationship. Starting with $\Lambda$, we construct algebras with finite…

Let $U_q(\hat{\cal G})$ be a quantized affine Lie algebra. It is proven that the universal R-matrix $R$ of $U_q(\hat{\cal G})$ satisfies the celebrated conjugation relation $R^\dagger=TR$ with $T$ the usual twist map. As applications, braid…

We study sun-products on $\R^n$, i.e. generalized Abelian deformations associated with star-products for general Poisson structures on $\R^n$. We show that their cochains are given by differential operators. As a consequence, the weak…

The moduli space of generalized deformations of a Calabi-Yau hypersurface is computed in terms of the Jacobian ring of the defining polynomial. The fibers of the tangent bundle to this moduli space carry algebra structures, which are…

Let A be an augmented algebra over a semi-simple algebra S. We show that the Ext algebra of S as an A-module, enriched with its natural A-infinity structure, can be used to reconstruct the completion of A at the augmentation ideal. We use…

The study of global deformations of Lie algebras is related to the problem of classification of simple Lie algebras over fields of small characteristic. The classification of finite-dimensional simple Lie algebras is complete over…

We study $n$-ary commutative superalgebras and $L_{\infty}$-algebras that possess a skew-symmetric invariant form, using the derived bracket formalism. This class of superalgebras includes for instance Lie algebras and their $n$-ary…

We construct a new family of infinite-dimensional quasi-graded Lie algebras on hyperelliptic curves. We show that constructed algebras possess infinite number of invariant functions and admit a decomposition into the direct sum of two…

This paper is devoted to deformation theory of graded Lie algebras over $\Z$ or $\Z_l$ with finite dimensional graded pieces. Such deformation problems naturally appear in number theory. In the first part of the paper, we use Schlessinger…

In this paper we generalize classical results on Lie algebras and universal enveloping algebras of Lie algebras to Lie-Rinehart algebras. We define for any Lie-Rinehart algebra $L$ and any cocycle $f$ in $Z^2(L,B)$, a universal enveloping…

Let $k$ be an arbitrary field and $d$ a positive integer. For each degenerate symmetric or antisymmetric bilinear form $M$ on $k^{d}$ we determine the structure of the Lie algebra of matrices that preserve $M$, and of the Lie algebra of…