Related papers: Weak faces of highest weight modules and root syst…

Let $G={\rm GL}_n$ be the general linear group over an algebraically closed field $k$, let $\mathfrak g=\mathfrak gl_n$ be its Lie algebra and let $U$ be the subgroup of $G$ which consists of the upper uni-triangular matrices. Let…

Let $\Phi$ be a finite crystallographic irreducible root system and $\mathcal P_{\Phi}$ be the convex hull of the roots in $\Phi$. We give a uniform explicit description of the polytope $\mathcal P_{\Phi}$, analyze the…

The principal admissible representations of affine Kac-Moody algebras are studied, with a view to their use in conformal field theory. We discuss the generation of the set of principal admissible highest weights, concentrating mainly on…

The aim of this work is to study a very special family of odd-quadratic Lie superalgebras ${\mathfrak g}={\mathfrak g}_{\bar 0}\oplus {\mathfrak g}_{\bar 1}$ such that ${\mathfrak g}_{\bar 1}$ is a weak filiform ${\mathfrak g}_{\bar…

Generalizing the super duality formalism for finite-dimensional Lie superalgebras of type $ABCD$, we establish an equivalence between parabolic BGG categories of a Kac-Moody Lie superalgebra and a Kac-Moody Lie algebra. The characters for a…

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…

Given a map $\Xi\colon U(\mathfrak{g})\rightarrow A$ of associative algebras, with $U(\mathfrak{g})$ the universal enveloping algebra of a (complex) finite-dimensional reductive Lie algebra $\mathfrak{g}$, the restriction functor from…

We introduce and study a new class of representations of surface groups into Lie groups of Hermitian type, called weakly maximal representations. They are defined in terms of invariants in bounded cohomology and extend considerably the…

Let $\mathfrak{g}$ be a semisimple Lie algebra over $\mathbb{C}$ having rank $l$ and let $V=L(\lambda)$ be an irreducible finite-dimensional $\mathfrak{g}$-module having highest weight $\lambda.$ Computations of weight multiplicities in…

We give a presentation of localized affine and degenerate affine Hecke algebras of arbitrary type in terms of weights of the polynomial subalgebra and varied Demazure-BGG type operators. We offer a definition of a graded algebra…

We classify all irreducible highest-weight unitary modules over the non-compact real form $\mathfrak{u}(p,q|n)$ of the general linear Lie superalgebra $\mathfrak{gl}_{p+q|n}$. The classification is given by explicit necessary and sufficient…

We classify Jet modules for the Lie (super)algebras $\mathfrak{L}=W\ltimes(\mathfrak{g}\otimes\mathbb{C}[t,t^{-1}])$, where $W$ is the Witt algebra and $\mathfrak{g}$ is a Lie superalgebra with an even diagonlizable derivation. Then we give…

Let G be a connected reductive algebraic group and H be a reductive closed and connected subgroup of G both defined on an algebraically closed field of characteristic zero. We consider the set C of the couple (x,y) of the dominant weights…

Let $\mathfrak{g}$ be a complex simple Lie algebra. A simple $\mathfrak{g}$-module is called minimal if the associated variety of its annihilator ideal coincides with the closure of the minimal nilpotent coadjoint orbit. The main result of…

We introduce and study "2-roots", which are symmetrized tensor products of orthogonal roots of Kac--Moody algebras. We concentrate on the case where $W$ is the Weyl group of a simply laced Y-shaped Dynkin diagram $Y_{a,b,c}$ having $n$…

We consider the principal subspaces of certain level $k\geqslant 1$ integrable highest weight modules and generalized Verma modules for the untwisted affine Lie algebras in types $D$, $E$ and $F$. Generalizing the approach of G. Georgiev we…

We define global and local Weyl modules for $q \otimes A$, where $q$ is the queer Lie superalgebra and $A$ is an associative commutative unital $\mathbb{C}-$algebra. We prove that global Weyl modules are universal highest weight objects in…

Let $\mathfrak g = \mathfrak{gl}_N(k)$, where $k$ is an algebraically closed field of characteristic $p > 0$, and $N \in \mathbb Z_{\ge 1}$. Let $\chi \in \mathfrak g^*$ and denote by $U_\chi(\mathfrak g)$ the corresponding reduced…

We present an invariant of connected and oriented closed 3-manifolds based on a coribbon Weak Hopf Algebra H with a suitable left-integral. Our invariant can be understood as the generalization to Weak Hopf Algebras of the…

Weyl-orbit functions have been defined for each simple Lie algebra, and permit Fourier-like analysis on the fundamental region of the corresponding affine Weyl group. They have also been discretized, using a refinement of the coweight…