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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…
In this paper all of the classical constructions of A. Young are generalized to affine Hecke algebras of type A. It is proved that the calibrated irreducible representations of the affine Hecke algebra are indexed by placed skew shapes and…
The notion of $q$-grading on the enveloping algebra generated by products of q-deformed Heisenberg algebras is introduced for $q$ complex number in the unit disc. Within this formulation, we consider the extension of the notion of…
Let G be a reductive algebraic group over a local field K or a global field F. It is well know that there exists a non-trivial and interesting representation theory of the group G(K) as well as the theory of automorphic forms on the…
In this thesis we study algebraic structures in M-theory, in particular the exceptional Lie algebras arising in dimensional reduction of its low energy limit, eleven-dimensional supergravity. We focus on e8 and its infinite-dimensional…
The structure of the general, inhomogeneous solution of (bosonic) Einstein-matter systems in the vicinity of a cosmological singularity is considered. We review the proof (based on ideas of Belinskii-Khalatnikov-Lifshitz and technically…
We investigate the representation theory of the rational and trigonometric Cherednik algebra of type $GL_n$ by means of combinatorics on periodic (or cylindrical) skew diagrams. We introduce and study standard tableaux and plane partitions…
We construct the vertex operator representation for the Affine Kac-Moody $SL(M+K+1)$ algebra, which is relevant for the construction of the soliton solutions of the constrained KP hierarchies. The oscillators involved in the vertex operator…
Given a finite group $G$ and a subgroup $K$, we study the commutant of $\text{Ind}_K^G\theta$, where $\theta$ is an irreducible $K$-representation. After a careful analysis of Frobenius reciprocity, we are able to introduce an orthogonal…
Geroch's solution-generating method is extended to the case of Einstein spaces, which possess a Killing vector {{}and are thus asymptotically (locally) (anti-)de Sitter}. This includes the reduction to a three-dimensional coset space, the…
We survey some of the AGT relations between N=2 gauge theories in four dimensions and geometric representations of symmetry algebras of two-dimensional conformal field theory on the equivariant cohomology of their instanton moduli spaces.…
The reduction of ten-dimensional heterotic supergravity with Yang-Mills symmetry group K is performed on an arbitrary n-dimensional group manifold G. The reduction involves a nonvanishing 3-form flux, and the Lie algebra of G must have…
We review the recently discovered interplay between chaos and symmetry in the general inhomogeneous solution of many string-related Einstein-matter systems in the vicinity of a cosmological singularity. The…
We give explicit constructions of some finite-dimensional representations of generalized double affine Hecke algebras (GDAHA) of higher rank using $R$-matrices for $U_q(\mathfrak{sl}_N)$. Our construction is motivated by an analogous…
Let the term $k$-representation refer to the permutation representations of the symmetric group $\mathfrak{S}_n$ on $k$-tuples and $k$-subsets as well as the $S^{(n-k,1^k)}$ irreducible representation of $\mathfrak{S}_n$. Endow…
We review the recently constructed non-trivial fermionic representations of the infinite-dimensional subalgebra K(E10) of the hyperbolic Kac--Moody algebra E10. These representations are all unfaithful (and more specifically, of finite…
In this paper we consider the representation theory of N=1 Super-W-algebras with two generators for conformal dimension of the additional superprimary field between two and six. In the superminimal case our results coincide with the…
These lectures given in Montreal in Summer 1997 are mainly based on, and form a condensed survey of, the book by N. Chriss and V. Ginzburg: `Representation Theory and Complex Geometry', Birkhauser 1997. Various algebras arising naturally in…
Recently a new class of quantum integrable models, the cyclotomic Gaudin models, were described in arXiv:1409.6937, arXiv:1410.7664. Motivated by these, we identify a class of affine hyperplane arrangements that we call cyclotomic…
We introduce a new class of infinite-dimensional Lie algebras, which we refer to as continuum Kac-Moody algebras. Their construction is closely related to that of usual Kac-Moody algebras, but they feature a continuum root system with no…