Related papers: A general approach to constructing minimal represe…
In the recent paper [AF12], we introduced an analysis of the Brylinski-Kostant model for spherical minimal representations for simple real Lie groups of non Hermitian type. We generalize here that analysis and give a unified geometric…
We construct explicit Drinfel'd twists of Jordanian type for the generalized Cartan type K Lie algebras in characteristic 0 and obtain the corresponding quantizations, especially their integral forms. By making modular reductions including…
This paper completes a series devoted to explicit constructions of finite-dimensional irreducible representations of the classical Lie algebras. Here the case of odd orthogonal Lie algebras (of type B) is considered (two previous papers…
In this paper we construct an "abstract Fock space" for general Lie types that serves as a generalisation of the infinite wedge $q$-Fock space familiar in type $A$. Specifically, for each positive integer $\ell$, we define a…
We consider Picard surfaces, locally symmetric varieties $S_{\Gamma}$ attached to the Lie group SU(2,1), and we construct explicit differential forms on $S_{\Gamma}$ representing Eisenstein classes, i.e. cohomology classes restricting…
We describe a new approach for classifying conjugacy classes of elementary abelian subgroups in simple algebraic groups over an algebraically closed field, and understanding the normaliser and centraliser structure of these. For toral…
We construct a Bernstein-Gelfand-Gelfand type resolution in terms of direct sums of Kac modules for the finite-dimensional irreducible tensor representations of the general linear superalgebra. As a consequence it follows that the unique…
Motivated by bases of representations compatible with the PBW filtration for basic Lie superalgebras by Kus and Fourier, we generalise the construction of degenerations of flag varieties via favourable modules to the super setup. In the…
We study the representation theory of the Lie superalgebra $\mathfrak{gl}(1|1)$, constructing two spectral sequences which eventually annihilate precisely the superdimension zero indecomposable modules in the finite-dimensional category.…
In this note, we formulate and prove branching rules of simple polynomial modules for the Lie superalgebra $\mathfrak{gl}(m|n)$. Our branching rules depend on the conjugacy class of the Borel subalgebra. A Gelfand-Tsetlin basis of a…
We classify simple differential Lie and Jordan (super)coalgebras of finite rank. In particular, we provide an explicit description of the Lie supercoalgebras associated with the operator product expansion (OPE) of the n=2,3,4 superconformal…
The classical Tits construction provides models of the exceptional simple Lie algebras in terms of a unital composition algebra and a degree three simple Jordan algebra. A couple of actions of the symmetric group of degree 4 on this…
A class of non-semisimple extensions of Lie superalgebras is studied. They are obtained by adjoining to the superalgebra its adjoint representation as an abelian ideal. When the superalgebra is of affine Kac-Moody type, a generalisation of…
In this paper, we develop the foundations of the representation theory of quiver Hecke--Clifford superalgebras. We further construct a Schur--Weyl duality between quantum affine analogues of the queer Lie superalgebra and the quiver…
We investigate the representations and the structure of Hecke algebras associated to certain finite complex reflection groups. We first describe computational methods for the construction of irreducible representations of these algebras,…
One of the algebraic structures that has emerged recently in the study of the operator product expansions of chiral fields in conformal field theory is that of a Lie conformal algebra [K]. A Lie pseudoalgebra is a generalization of the…
We explicitly construct a particular real form of the Lie algebra $\mathfrak{e}_7$ in terms of symplectic matrices over the octonions, thus justifying the identifications $\mathfrak{e}_7\cong\mathfrak{sp}(6,\mathbb{O})$ and, at the group…
Given any representation of an arbitrary Lie algebra L over a field k of characteristic 0, we construct representations of L on bosonic and fermionic Fock space. The method gives an explicit formula for a (sometimes trivial) 2-cocycle in…
The present paper deals with the representation theory of the reflection equation algebra, connected with a Hecke type R-matrix. Up to some reasonable additional conditions the R-matrix is arbitrary (not necessary originated from quantum…
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…