Related papers: Defining an Affine Partition Algebra

This article contains a review of categorifications of semisimple representations of various rings via abelian categories and exact endofunctors on them. A simple definition of an abelian categorification is presented and illustrated with…

We develop new techniques to classify basic algebras of blocks of finite groups over algebraically closed fields of prime characteristic. We apply these techniques to simplify and extend previous classifications by Linckelmann, Murphy and…

We introduce an analogue of the $q$-Schur algebra associated to Coxeter systems of type $\hat A_{n-1}$. We give two constructions of this algebra. The first construction realizes the algebra as a certain endomorphism algebra arising from an…

An affine Hecke algebras can be realized as an equivariant K-group of the corresponding Steinberg variety. This gives rise naturally to some two-sided ideals of the affine Hecke algebra by means of the closures of nilpotent orbits of the…

New realizations of finite W algebras are constructed by relaxing the usual constraint conditions. Then, finite W algebras are recognized in the Heisenberg quantization recently proposed by Leinaas and Myrheim, for a system of two identical…

We classify, up to isomorphism and up to equivalence, division gradings (by abelian groups) on finite-dimensional simple real algebras. Gradings on finite-dimensional simple algebras are determined by division gradings, so our results give…

In this article, we define and study a geometry and an order on the set of partitions of an even number of objects. One of the definitions involves the partition algebra, a structure of algebra on the set of such partitions depending on an…

This paper is a sequel to arXiv:2307.13358 and arXiv:2308.16090. A construction associating a semialgebra with an algebra, subalgebra, and a coalgebra dual to the subalgebra played a central role in the author's book arXiv:0708.3398. In…

We develop filtered-graded techniques for algebras in monoidal categories with the main goal of establishing a categorical version of Bongale's 1967 result: A filtered deformation of a Frobenius algebra over a field is Frobenius as well.…

We prove a categorical duality between a class of abstract algebras of partial functions and a class of (small) topological categories. The algebras are the isomorphs of collections of partial functions closed under the operations of…

Affine difference algebraic groups are a generalization of affine algebraic groups obtained by replacing algebraic equations with algebraic difference equations. We show that the isomorphism theorems from abstract group theory have…

For any finitely generated abelian group $Q$, we reduce the problem of classification of $Q$-graded simple Lie algebras over an algebraically closed field of "good" characteristic to the problem of classification of gradings on simple Lie…

An algebra is said to be \emph{$\tau$-tilting finite} provided it has only a finite number of $\tau$-rigid objects up to isomorphism. We associate a category to each such algebra. The objects are the wide subcategories of its category of…

Affine structures on a Lie groupoid, including affine $k$-vector fields, $k$-forms and $(p,q)$-tensors are studied. We show that the space of affine structures is a 2-vector space over the space of multiplicative structures. Moreover, the…

Schur--Weyl--Jones duality establishes the connection between the commuting actions of the symmetric group $S_{n}$ and the partition algebra $P_{k}(n)$ on the tensor space $\left(\mathbb{C}^n\right)^{\otimes k}.$ We give a refinement of…

This is the second in a series of papers laying the foundations for a differential graded approach to derived differential geometry (and other geometries in characteristic zero). In this paper, we extend the classical notion of a dg-algebra…

Let $X$ be a nonempty set and $X^{2}$ be the Cartesian square of $X$. Some semigroups of binary relations generated partitions of $X^2$ are studied. In particular, the algebraic structure of semigroups generated by the finest partition of…

In this paper, the complete algebraic structure of finite semisimple group algebra of a normally monomial group is described. The main result is illustrated by computing the explicit Wedderburn decomposition of finite semisimple group…

Chevalley's theorem states that every smooth connected algebraic group over a perfect field is an extension of an abelian variety by a smooth connected affine group. That fails when the base field is not perfect. We define a pseudo-abelian…

It is known that the notion of graded differential algebra coincides with the notion of monoid in the monoidal category of complexes. By using the monoidal structure introduced by M. Kapranov for the category of $N$-complexes we define the…