Related papers: Finite dimensional simple modules of $(q, \mathbf{…

Let $K$ be a fixed field. We attach to each column-finite quiver $E$ a von Neumann regular $K$-algebra $Q(E)$ in a functorial way. The algebra $Q(E)$ is a universal localization of the usual path algebra $P(E)$ associated with $E$. The…

Let $\mathcal{B}_{\mathfrak{q}}$ be a finite-dimensional Nichols algebra of diagonal type corresponding to a matrix $\mathfrak{q} \in \mathbf{k}^{\theta \times \theta}$, where $\mathbf{k}$ is an algebraically closed field of characteristic…

This paper is a short account of the construction of a new class of the infinite-dimensional representations of the quantum groups. The examples include finite-dimensional quantum groups $U_q(\mathfrak{g})$, Yangian $Y(\mathfrak{g})$ and…

When the parameter $q\in\mathbb C^*$ is not a root of unity, simple modules of affine $q$-Schur algebras have been classified in terms of Frenkel--Mukhin's dominant Drinfeld polynomials (\cite[4.6.8]{DDF}). We compute these Drinfeld…

We formulate a $q$-Schur algebra associated to an arbitrary $W$-invariant finite set $X_{\texttt f}$ of integral weights for a complex simple Lie algebra with Weyl group $W$. We establish a $q$-Schur duality between the $q$-Schur algebra…

We present some recently discovered infinite dimensional Lie algebras that can be understood as extensions of the algebra Map(M,g) of maps from a compact p-dimensional manifold to some finite dimensional Lie algebra g. In the first part of…

Let $X$ and $\mathfrak{a}$ be an affine scheme and (respectively) a finite-dimensional associative algebra over an algebraically-closed field $\Bbbk$, both equipped with actions by a linearly-reductive linear algebraic group $G$. We…

For quantized universal enveloping algebras we construct weight modules by inducing representations of the centralizer of the Cartan subalgebra in the quantized universal enveloping algebra. The induced modules arising from…

We investigate the characters of some finite-dimensional representations of the quantum affine algebras $U_q(\hat{g})$ using the action of the copy of $U_q(g)$ embedded in it. First, we present an efficient algorithm for computing the…

Motivated by the theory of unitary representations of finite dimensional Lie supergroups, we describe those Lie superalgebras which have a faithful finite dimensional unitary representation. We call these Lie superalgebras unitary. This is…

Let $\mathbb F$ denote a field, and pick a nonzero $q \in \mathbb F$ that is not a root of unity. Let $\mathbb Z_4=\mathbb Z/4 \mathbb Z$ denote the cyclic group of order 4. Define a unital associative ${\mathbb F}$-algebra $\square_q$ by…

We construct infinite-dimensional analogues of finite-dimensional simple modules of the nonstandard $q$-deformed enveloping algebra $U_q'(\mathfrak{so}_n)$ defined by Gavrilik and Klimyk, and we do the same for the classical universal…

We review some important facts about the structure of tensor products of finite dimensional representations of quantum affine algebras. This is done from the elementary standpoint of the representation theory of semisimple Lie algebras in…

Level-one representations of the quantum affine superalgebra $U_q[\hat{gl(N|N)}]$ associated to the appropriate non-standard system of simple roots and $q$-vertex operators (intertwining operators) associated with the level-one modules are…

The aim of this paper is to give a complete classification of irreducible finite dimensional representations of the nonstandard q-deformation U'_q(so(n)) (which does not coincide with the Drinfeld-Jimbo quantum algebra U_q(so(n)) of the…

A complete list of Uq(sl2)-module algebra structures on the quantum plane is produced and the (uncountable family of) isomorphism classes of these structures are described. The composition series of representations in question are computed.…

We describe the generators and prove a number of relations for the construction of a planar algebra from the restricted quantum group $\bar{U}_{q}(\mathfrak{sl}_{2})$. This is a diagrammatic description of…

We develop the basic formalism of complex $q$-analysis to study the solutions of second order $q$-difference equations which reduce, in the $q\rightarrow 1$ limit, to the ordinary Laplace equation in Euclidean and Minkowski space. After…

Let k be a field, Q a finite directed graph, and kQ its path algebra. Make kQ an N-graded algebra by assigning each arrow a positive degree. Let I be an ideal in kQ generated by a finite number of paths and write A = kQ/I. Let QGr A denote…

The construction approach proposed in the previous paper Ref. 1 allows us there and in the present paper to construct at generic deformation parameter $q$ all finite--dimensional representations of the quantum Lie superalgebra…