Related papers: Quantum Kac-Moody Algebras and Vertex Representati…

In conformal field theories, when the conformal symmetry is enhanced by a global Lie group symmetry, the original Virasoro algebra can be extended to Kac-Moody algebra. In this paper, we extend the lattice construction of the Kac-Moody…

We study representations of the Loop Kac-Moody Lie algebra g \otimes A, where g is any Kac-Moody algebra and A is a ring of Laurent polynomials in n commuting variables. In particular, we study representations with finite dimensional weight…

We report on recent work concerning a new type of generalised Kac-Moody algebras based on the spaces of differentiable mappings from compact manifolds or homogeneous spaces onto compact Lie groups.

We categorify the highest weight integrable representations and their tensor products of a symmetric quantum Kac-Moody algebra. As byproducts, we obtain a geometric realization of Lusztig's canonical bases of these representations as well…

The present paper develops a general theory of quantum group analogs of symmetric pairs for involutive automorphism of the second kind of symmetrizable Kac-Moody algebras. The resulting quantum symmetric pairs are right coideal subalgebras…

We pursue our study of generalised Kac-Moody and Virasoro algebras defined on compact homogeneous manifolds. Extending the well-known Vertex operator in the case of the two-torus or the two-sphere, we obtain explicit bosonic realisations of…

Borcherds algebras represent a new class of Lie algebras which have almost all the properties that ordinary Kac-Moody algebras have, and the only major difference is that these generalized Kac-Moody algebras are allowed to have imaginary…

We consider a natural generalisation of the class of hyperbolic Kac-Moody algebras. We describe in detail the conditions under which these algebras are Lorentzian. We also construct their fundamental weights, and analyse whether they…

We construct a 2-category associated with a Kac-Moody algebra and we study its 2-representations. This generalizes earlier work with Chuang for type A. We relate categorifications relying on K_0 properties and 2-representations.

Continuum Kac-Moody algebras have been recently introduced by the authors and O. Schiffmann. These are Lie algebras governed by a continuum root system, which can be realized as uncountable colimits of Borcherds-Kac-Moody algebras. In this…

In this paper we generalize Drinfeld's twisted quantum affine algebras to construct twisted quantum algebras for all simply-laced generalized Cartan matrices and present their vertex representation realizations.

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 consider the problem of representing the Kac-Moody algebra $\mathfrak{g}(N)$ specified by an $r\times r$ indecomposable generalised Cartan matrix $N$ as vector fields on the torus ${{\bb C}^*}^r$. It is shown that, if the representations…

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

We introduce a generalization of the classical Hall-Littlewood and Kostka-Foulkes polynomials to all symmetrizable Kac-Moody algebras. We prove that these Kostka-Foulkes polynomials coincide with the natural generalization of Lusztig's…

This paper deals with the problem of conjugacy of Cartan subalgebras for affine Kac-Moody Lie algebras. Unlike the methods used by Peterson and Kac, our approach is entirely cohomological and geometric. It is deeply rooted on the theory of…

Let $C$ be a symmetrizable generalized Cartan Matrix, and $q$ an indeterminate. ${\fg}(C)$ is the Kac-Moody Lie algebra and $U=U_q({\fg}(C))$ the associated quantum enveloping algebra over $ k={\Bbb Q}(q)$. The quantum function algebra…

We study the canonical basis for the negative part of the quantum generalized Kac-Moody algebra associated to a symmetric Borcherds-Cartan matrix. The algebras associated to two different matrices satisfying certain conditions may coincide.…

We consider quiver representations respecting a quiver automorphism and show that the dimension vectors of the indecomposables are precisely the positive roots of an associated symmetrisable Kac-Moody Lie algebra. Moreover, every such Lie…

The method of graded contractions, based on the preservation of the automorphisms of finite order, is applied to the affine Kac-Moody algebras and their representations, to yield a new class of infinite dimensional Lie algebras and…