Irreducible representations of quantum solvable algebras at roots of 1

This text provides an introduction and complements to some basic constructions and results in 2-representation theory of Kac-Moody algebras.

The goals of this article are as follows: (1) To determine the irreducible components of the affine varieties parametrizing the representations of $ \Lambda $ with dimension vector d, where $ \Lambda $ traces a major class of finite dimensional algebras; (2) To generically describe the representations encoded by the components. The target class consists of those truncated path algebras $ \Lambda $ over an algebraically closed field K which are based on a quiver Q without orie...

We classify the centers of the quantized Weyl algebras that are PI and derive explicit formulas for the discriminants of these algebras over a general class of polynomial central subalgebras. Two different approaches to these formulas are given: one based on Poisson geometry and deformation theory, and the other using techniques from quantum cluster algebras. Furthermore, we classify the PI quantized Weyl algebras that are free over their centers and prove that their discrimi...

We construct the quantized enveloping algebra of any simple Lie algebra of type ADE as the quotient of a Grothendieck ring arising from certain cyclic quiver varieties. In particular, the dual canonical basis of a one-half quantum group with respect to Lusztig's bilinear form is contained in the natural basis of the Grothendieck ring up to rescaling. This paper expands the categorification established by Hernandez and Leclerc to the whole quantum groups. It can be viewed as...

In this paper we study the representation dimension as well as the derived dimension of the path algebra of an artin algebra over a finite and acyclic quiver.

Let $\mathfrak{g}$ be a solvable Lie algebra and $Q$ an $ad \mathfrak{g}$-stable prime ideal of the symmetric algebra $S(\mathfrak{g})$ of $\mathfrak{g}$. If $E$ is the set of non zero elements of $S(\mathfrak{g})/Q$ which are eigenvectors for the adjoint action of $\mathfrak{g}$ in $S(\mathfrak{g})/Q$, the localised algebra $(S(\mathfrak{g})/Q)_{E}$ has a natural structure of Poisson algebra. We study this algebra here.

We consider various specializations of the non-twisted quantum affine algebras at roots of unity. We define and study the q-characters of their finite-dimensional representations.

This review article discusses recent progress in understanding of various families of integrable models in terms of algebraic geometry, representation theory, and physics. In particular, we address the connections between soluble many-body systems of Calogero-Ruijsenaars type, quantum spin chains, spaces of opers, representations of double affine Hecke algebras, enumerative counts to quiver varieties, to name just a few. We formulate several conjectures and open problems. Thi...

This note is a development of our two previous papers, arXiv:1212.3392v1 and 1306.3660v1. The fundamental question is whether there exists a Galois theory, in which the Galois group is a quantum group. For a linear equations with respect to a Hopf algebra, we arrived at a final form if the base field consists of constants. In this case, we have non-commutative Picard-Vessiot rings and asymmetric Tannaka theory. For non-linear equations there are examples that might make...

A very elementary introduction to quantum algebras is presented and a few examples of their physical applications are mentioned.