Related papers: Asssociative algebras under multi-commutator
In this paper we consider all these nonassociative algebras defined by the action of invariant subspaces of the symmetric group $\Sigma_3$ on the associator of the considered laws.
In this article, we mainly study the products of commutator ideals of Lie-admissible algebras such as Novikov algebras, bicommutative algebras, and assosymmetric algebras. More precisely, we first study the properties of the lower central…
A new 3-ary non-associative algebra, which is called a semi-associative $3$-algebra, is introduced, and the double modules and double extensions by cocycles are provided. Every semi-associative $3$-algebra $(A, \{ , , \})$ has an adjacent…
We define the notion of an almost polynomial identity of an associative algebra $R$, and show that its existence implies the existence of an actual polynomial identity of $R$. A similar result is also obtained for Lie algebras and Jordan…
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 $\HH^{\ast}(A)$…
Over a field of characteristic zero, we show that two commutative differential graded (dg) algebras are quasi-isomorphic if and only if they are quasi-isomorphic as associative dg algebras. This answers a folklore problem in rational…
For a couple of associative algebras we define the notion of their double and give a set of examples. Also, we discuss applications of such doubles to representation theory of certain quantum algebras and to a new type of Noncommutative…
We set up a homological algebra for N-complexes, which are graded modules together with a degree -1 endomorphism d satisfying d^N=0. We define Tor- and Ext-groups for N-complexes and we compute them in terms of their classical counterparts…
A skew Calabi-Yau algebra is a generalization of a Calabi-Yau algebra which allows for a non-trivial Nakayama automorphism. We prove three homological identities about the Nakayama automorphism and give several applications. The identities…
An algebra $L$ over a field $\Bbb F$, in which product is denoted by $[\,,\,]$, is said to be \textit{ Lie type algebra} if for all elements $a,b,c\in L$ there exist $\alpha, \beta\in \Bbb F$ such that $\alpha\neq 0$ and $[[a,b],c]=\alpha…
Associated to a symmetric space there is a canonical connection with zero torsion and parallel curvature. This connection acts as a binary operator on the vector space of smooth sections of the tangent bundle, and it is linear with respect…
The symmetric homology of a unital algebra $A$ over a commutative ground ring $k$ is defined using derived functors and the symmetric bar construction of Fiedorowicz. For a group ring $A = k[\Gamma]$, the symmetric homology is related to…
We introduce two remarkable identities written in terms of single commutators and anticommutators for any three elements of arbitrary associative algebra. One is a consequence of other (fundamental identity). From the fundamental identity,…
Hom-algebras are generalizations of algebras obtained using a twisting by a linear map. But there is a priori a freedom on where to twist. We enumerate here all the possible choices in the Lie and associative categories and study the…
It is shown that commutator identities on associative algebras generate solutions of linearized integrable equations. Next, a special kind of the dressing procedure is suggested that in a special class of integral operators enables to…
Motivated by the combinatorial properties of products in Lie algebras, we investigate the subset of permutations that naturally appears when we write the long commutator $[x_1, x_2, ..., x_m]$ as a sum of associative monomials. We…
We review the notion of submanifold algebra, as introduced by T. Masson, and discuss some properties and examples. A submanifold algebra of an associative algebra $A$ is a quotient algebra $B$ such that all derivations of $B$ can be lifted…
The main non-associative algebras are Lie algebras and Jordan algebras. There are several ways to unify these non-associative algebras and associative algebras.
From the symmetry between definitions of left and right divisors in associative $D$-algebra $A$, the possibility to define quotient as $A\otimes A$-number follows. In the paper, I considered division and division with remainder. I…
In this paper, we develop a method to obtain the algebraic classification of compatible pre-Lie algebras from the classification of pre-Lie algebras of the same dimension. We use this method to obtain the algebraic classification of complex…