Related papers: The Biderivative and A_\infty-bialgebras
Binary operations on algebras of observables are studied in the quantum as well as in the classical case. It is shown that certain natural compatibility conditions with the associative product imply the properties which usually are…
We introduce a notion of pre-alternative algebra which may be seen as an alternative algebra whose product can be decomposed into two pieces which are compatible in a certain way. It is also the "alternative" analogue of a dendriform…
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…
A subalgebra $A$ of the algebra $B(\mathcal{H})$ of bounded linear operators on a separable Hilbert space $\mathcal{H}$ is said to be catalytic if every transitive subalgebra $\mathcal{T}\subset B(\mathcal{H})$ containing it is strongly…
An averaging operator on an associative algebra $A$ is an algebraic abstraction of the time average operator on the space of real-valued functions defined in time-space. In this paper, we consider relative averaging operators on a bimodule…
For a Lie-Rinehart algebra (A,L), generators for the Gerstenhaber algebra \Lambda_A L correspond bijectively to right (A,L)-connections on A in such a way that B-V structures correspond to right (A,L)-module structures on A. When L is…
Given two baric algebras $(A_1,\omega_1)$ and $(A_2,\omega_2)$ we describe a way to define a new baric algebra structure over the vector space $A_1\oplus A_2$, which we shall denote $(A_1\bowtie A_2,\omega_1\bowtie\omega_2)$. We present…
We classify all 2-term $L_\infty$-algebras up to isomorphism. We show that such $L_\infty$-algebras are classified by a Lie algebra, a vector space, a representation (all up to isomorphism) and a cohomology class of the corresponding Lie…
We endow the category of bialgebras over a pair of operads in distribution with a cofibrantly generated model category structure. We work in the category of chain complexes over a field of characteristic zero. We split our construction in…
We present a study of the homological algebra of bimodules over $A_\infty$-algebras endowed with an involution. Furthermore we introduce a derived description of Hochschild homology and cohomology for involutive $A_\infty$-algebras.
An equivalence between Lu's bialgebroids, Xu's bialgebroids with an anchor and Takeuchi's $\times_{A}$-bialgebras is explicitly proven. A new class of examples of bialgebroids is constructed. A (formal) dual of a bialgebroid, termed…
We present an elementary and self-contained construction of $A_\infty$-algebras, $A_\infty$-bimodules and their Hochschild homology and cohomology groups. In addition, we discuss the cup product in Hochschild cohomology and the spectral…
We study in this article the concepts of algebra up to homotopy for a structure defined by two operations $ \pt $ and $[, ]$. Having determined the structure of $ G_\infty $ algebras and $ P_\infty $ algebras, we generalize this…
We show that the tensor product of two cyclic $A_\infty$-algebras is, in general, not a cyclic $A_\infty$-algebra, but an $A_\infty$-algebra with homotopy inner product. More precisely, we construct an explicit combinatorial diagonal on the…
We introduce the notion of an $L_{\infty}$-bialgebra structure on a vector space. We show that the rational homotopy groups $\pi_{*}(\Omega \Sigma Y)\otimes \mathbb{Q}$ admit such a structure for the loop space $\Omega \Sigma Y$ of a…
We study Hom-bialgebras and objects admitting coactions by Hom-bialgebras. In particular, we construct a Hom-bialgebra M representing the functor of 2x2-matrices on Hom-associative algebras. Then we construct a Hom-algebra analogue of the…
The dihedral homology functor $HD:A_\infty^{{\rm inv}}(K)\to GrM(K)$ from the category $A_\infty^{{\rm inv}}(K)$ of involutive $A_\infty$-algebras over any commutative unital ring $K$ to the category $GrM(K)$ of graded $K$-modules is…
We establish a structure theorem, analogous to the classical result of Milnor and Moore, for differential graded Hopf algebras: any differential Hopf algebra $H$ that is free as a coalgebra carries an underlying $B_\infty$ algebra structure…
We study extensively the homotopy theory of coalgebras. By coalgebras, we mean the full theory of coalgebras: with counits and not necessarily locally conilpotent. For example $\mathcal E_\infty$-coalgebras, $\mathcal A_\infty$-coalgebras,…
A difference operator on an associative algebra is an algebraic abstraction of the forward and backward difference operators. In this paper, we first introduce difference operators on associative $2$-algebras and consider the category of…