Related papers: On the relation between integrability and infinite…
A finite W-algebra is an associative algebra constructed from a semisimple Lie algebra and its nilpotent element. In this survey we review recent developments in the representation theory of W-algebras. We emphasize various interactions…
We continue the study of quantum Liouville theory through Polyakov's functional integral \cite{Pol1,Pol2}, started in \cite{T1}. We derive the perturbation expansion for Schwinger's generating functional for connected multi-point…
We show that there exists a purely infinite AH-algebra. The AH-algebra arises as an inductive limit of C*-algebras of the form C_0([0,1),M_k) and it absorbs the Cuntz algebra O_\infty tensorially. Thus one can reach an O_\infty-absorbing…
Integrable structures arise in general relativity when the spacetime possesses a pair of commuting Killing vectors admitting 2-spaces orthogonal to the group orbits. The physical interpretation of such spacetimes depends on the norm of the…
Let $\fg$ be any untwisted affine Kac-Moody algebra, $\mu$ any fixed complex number, and $\wt\fg(\mu)$ the corresponding toroidal extended affine Lie algebra of nullity two. For any $k$-tuple $\bm{\lambda}=({\lambda}_1, \cdots,…
We review some basic features of the Lie-algebraic classification of W-algebras and related integrable hierarchies in 1+1 dimensions, pointing out the role of affine Lie algebras. We emphasize that the supersymmetric extensions of the above…
We describe recent work on preprojective algebras and moduli spaces of their representations. We give an analogue of Kac's Theorem, characterizing the dimension types of indecomposable coherent sheaves over weighted projective lines in…
In these lectures I consider the Hitchin integrable systems and their relations with the self-duality equations and the twisted super-symmetric Yang-Mills theory in four dimension follow Hitchin and Kapustin-Witten. I define the Symplectic…
We study classical integrability of the supersymmetric U(N) $\sigma$ model with the Wess-Zumino-Witten term on full and half plane. We demonstrate the existence of nonlocal conserved currents of the model and derive general recursion…
The purpose of this article is to develop an algebraic approach to the problem of integrable classification of differential-difference equations with one continuous and two discrete variables. As a classification criterion, we put forward…
We prove a general theorem for constructing integral quantum cluster algebras over ${\mathbb{Z}}[q^{\pm 1/2}]$, namely that under mild conditions the integral forms of quantum nilpotent algebras always possess integral quantum cluster…
We carry out in full generality and without fixing specific boundary conditions, the symmetry and charge analysis near a generic null surface for two and three dimensional (2d and 3d) gravity theories. In 2d and 3d there are respectively…
We study the relation between the quantum integrable systems derived from the dimer graphs and five dimensional $\mathcal{N}=1$ supersymmetric gauge theories on $S^1 \times \mathbb{R}^4$. We construct integrable systems based on new dimer…
W-algebras of finite type are certain finitely generated associative algebras closely related to the universal enveloping algebras of semisimple Lie algebras. In this paper we prove a conjecture of Premet that gives an almost complete…
Starting from noncommutative Fermi theory in two-dimensions, we construct a deformed Kac-Moody algebra between its vector and Chiral currents . The higher-order corrections to the deformed Kac-Moody algebra are explicitly calculated. We…
In this review, we present a general framework for the construction of Kac-Moody (KM) algebras associated to higher-dimensional manifolds. Starting from the classical case of loop algebras on the circle $\mathbb{S}^{1}$, we extend the…
We present a superconformally invariant and integrable model based on the twisted affine Kac-Moody superalgebra $\hat{osp(2|2)}^{(2)}$ which is the supersymmetrization of the purely bosonic conformal affine Liouville theory recently…
For any L-infinity algebra L, we construct an A-infinity structure on the space of symmetric tensors Sym*(L), which generalizes the classical universal enveloping for Lie algebras. Our construction is based on an invariant homotopy on a…
This is a proceedings article reviewing a recent combinatorial construction of the su(n) WZNW fusion ring by C. Stroppel and the author. It contains one novel aspect: the explicit derivation of an algorithm for the computation of fusion…
The tensionless limit of gauged WZW models arises when the level of the underlying Kac-Moody algebra assumes its critical value, equal to the dual Coxeter number, in which case the central charge of the Virasoro algebra becomes infinite. We…