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We introduce a semisimple tensor category $\mc{O}^{int}_q(m|n)$ of modules over an quantum ortho-symplectic superalgebra. It is a natural counterpart of the category of finitely dominated integrable modules over the quantum classical…

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 exhibit an adjunction between a category of abstract algebras of partial functions and a category of set quotients. The algebras are those atomic algebras representable as a collection of partial functions closed under relative…

Suppose that W is a finite, unitary, reflection group acting on the complex vector space V and X is a subspace of V. Define N to be the setwise stabilizer of X in W, Z to be the pointwise stabilizer, and C=N/Z. Then restriction defines a…

Let $S$ be a unital associative ring and $S[t;\sigma,\delta]$ be a skew polynomial ring, where $\sigma$ is an injective endomorphism of $S$ and $\delta$ a left $\sigma$-derivation. For each $f\in S[t;\sigma,\delta]$ of degree $m>1$ with a…

In this paper, Lie super-bialgebra structures on a class of generalized super $W$-algebra $\mathfrak{L}$ are investigated. By proving the first cohomology group of $\mathfrak{L}$ with coefficients in its adjoint tensor module is trivial,…

Let W be an irreducible finitely generated Coxeter group. The geometric representation of W in GL(V) provides a discrete embedding in the orthogonal group of the Tits form (the associated bilinear form of the Coxeter group). If the Tits…

We study a numerical semigroup ring as an algebra over another numerical semigroup ring. The complete intersection property of numerical semigroup algebras is investigated using factorizations of monomials into minimal ones. The goal is to…

We analyze the BGG Category $\mathcal{O}$ over a large class of generalized Weyl algebras (henceforth termed GWAs). Given such a "triangular" GWA for which Category $\mathcal{O}$ decomposes into a direct sum of subcategories, we study in…

Extending the Wedderburn-Artin theory of (classically) semisimple associative rings to the realm of topological rings with right linear topology, we show that the abelian category of left contramodules over such a ring is split…

Let $A$ be a finite dimensional associative $\mathbb{K}$-algebra over an algebraically closed field $\mathbb{K}$ of characteristic zero. To $A$, we can associate its basic form that is given by a quiver $Q = (Q_0, Q_1)$ with an admissible…

By Theorem~1.12 of the paper "A Class of Representations of Hecke Algebras", if $W$ is a Coxeter group whose proper parabolic subgroups are finite, and if the module of a finite $W$-digraph $\Gamma$ is isomorphic to the module of a…

Constructions are given of Noetherian maximal orders that are finitely presented algebras over a field K, defined by monomial relations. In order to do this, it is shown that the underlying homogeneous information determines the algebraic…

The irreducible representations of full support in the rational Cherednik category $\mathcal{O}_c(W)$ attached to a Coxeter group $W$ are in bijection with the irreducible representations of an associated Iwahori-Hecke algebra. Recent work…

We explain an elementary topological construction of the Springer representation on the homology of (topological) Springer fibers of types C and D in the case of nilpotent endomorphisms with two Jordan blocks. The Weyl group and component…

Diagram semigroups are interesting algebraic and combinatorial objects, several types of them originating from questions in computer science and in physics. Here we describe diagram semigroups in a general framework and extend our…

We describe the structure of bimodules (over finite dimensional algebras) which have the property that the functor of tensoring with such a bimodule sends any module to a projective module. The main result is that all such bimodules are…

Let $W_a$ be an affine Weyl group and $\eta:W_a\longrightarrow W_0$ be the natural projection to the corresponding finite Weyl group. We say that $w\in W_a$ has finite Coxeter part if $\eta(w)$ is conjugate to a Coxeter element of $W_0$.…

Let W be an irreducible, finitely generated Coxeter group. The geometric representation provides an discrete embedding in the orthogonal group of the so-called Tits form. One can look at the representation modulo the kernel of this form; we…

We construct a finite dimensional quiver algebra from the non-simply laced type $B$ Dynkin diagram, which we call the type $B$ zigzag algebra. This leads to a faithful categorical action of the type $B$ braid group $\mathcal{A}(B)$, acting…