Related papers: Actor of an alternative algebra

For an object $A$ of a category of interest $\mathbb{C}$ we construct the group with operations ${\mathfrak{B}}(A)$ and the semidirect product ${\mathfrak{B}}(A)\ltimes A$ and prove that there exists an actor of $A$ in $\mathbb{C}$ if and…

Working in the setting of ideally exact categories, we investigate the representability of actions of unital non-associative algebras over a field. We show that, in general, such categories fail to be action representable: for instance, the…

We study the categorical-algebraic condition that internal actions are weakly representable (WRA) in the context of varieties of (non-associative) algebras over a field. Our first aim is to give a complete characterization of action…

We consider inclusions of type $(P\otimes A)^G\subset(P\otimes B)^G$, where $G$ is a compact quantum group of Kac type acting on a ${\rm II}_1$ factor $P$, and on a Markov inclusion of finite dimensional $C^*$-algebras $A\subset B$. In the…

In this paper, we introduce a novel generalization of the classical property of algebras known as "being alternative," which we term "partially alternative." This new concept broadens the scope of alternative algebras, offering a fresh…

Recently V.H.L\'opez Sol\'is and I.Shestakov solved an old problem by N.Jacobson on describing of unital alternative algebras containing the $2\times 2$ matrix algebra $M_2$ as a unital subalgebra. Here we give another description of…

We define the completion of an associative algebra $A$ in a set $M=\{M_1,\dots,M_r\}$ of $r$ right $A$-modules in such a way that if $\mathfrak a\subseteq A$ is an ideal in a commutative ring $A$ the completion $A$ in the (right) module…

We show that a graded commutative algebra A with any square zero odd differential operator is a natural generalization of a Batalin-Vilkovisky algebra. While such an operator of order 2 defines a Gerstenhaber (Lie) algebra structure on A,…

We consider, for each exchange matrix B, a category of geometric cluster algebras over B and coefficient specializations between the cluster algebras. The category also depends on an underlying ring R, usually the integers, rationals, or…

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…

In a recent paper, motivated by the study of central extensions of associative algebras, G. Janelidze introduces the notion of weakly action representable category. In this paper, we show that the category of Leibniz algebras is weakly…

Absolute algebras are a new type of algebraic structures, endowed with a meaningful notion of infinite sums of operations without supposing any underlying topology. Opposite to the usual definition of operadic calculus, they are defined as…

We define contragredient Lie algebras in symmetric categories, generalizing the construction of Lie algebras of the form $\mathfrak{g}(A)$ for a Cartan matrix $A$ from the category of vector spaces to an arbitrary symmetric tensor category.…

We say that an algebraic group $G$ over a field is anti-affine if every regular function on $G$ is constant. We obtain a classification of these groups, with applications to the structure of algebraic groups in positive characteristics, and…

Let G be a group and let A be the algebra of complex functions on G with finite support. The product in G gives rise to a coproduct on A making it a multiplier Hopf algebra. In fact, because there exist integrals, we get an algebraic…

Let $G$ be a finite group. There is a standard theorem on the classification of $G$-equivariant finite dimensional simple commutative, associative, and Lie algebras (i.e., simple algebras of these types in the category of representations of…

The paper deals with the configuration of subalgebras in generic $n$-dimensional $k$-argument anticommutative algebras and ``regular'' anticommutative algebras.

We define a generalization $\mathfrak{G}$ of the Grassmann algebra $G$ which is well-behaved over arbitrary commutative rings $C$, even when $2$ is not invertible. In particular, this enables us to define a notion of superalgebras that does…

A central extension is a regular epimorphism in a Barr exact category $\mathscr{C}$ satisfying suitable conditions involving a given Birkhoff subcategory of $\mathscr{C}$ (joint work with G. M. Kelly, 1994). In this paper we take…

The universal enveloping algebra of any simple Lie algebra g contains a family of commutative subalgebras, called the quantum shift of argument subalgebras math.RT/0606380, math.QA/0612798. We prove that generically their action on…