相关论文: Gaudin Algebras, RSK and Calogero-Moser Cells in T…

The Lie algebroids are generalization of the Lie algebras. They arise, in particular, as a mathematical tool in investigations of dynamical systems with the first class constraints. Here we consider canonical symmetries of Hamiltonian…

Using the geometry of the associated Calogero-Moser space, R. Rouquier and the author have attached to any finite complex reflection group $W$ several notions (Calogero-Moser left, right or two-sided cells, Calogero-Moser cellular…

We study an N-body Calogero model in the S_N-symmetric subspace of the positive definite Fock space. We construct a new algebra of S_N-symmetric operators represented on the symmetric Fock space, and find a natural orthogonal basis by…

Let G be a simply connected semisimple compact Lie group with standard Poisson structure, K a closed Poisson-Lie subgroup, 0<q<1. We study a quantization C(G_q/K_q) of the algebra of continuous functions on G/K. Using results of Soibelman…

The understanding of the topology of the spectra of quantum Schubert cell algebras hinges on the description of their prime factors by ideals invariant under the maximal torus of the ambient Kac-Moody group. We give an explicit description…

The faces of the braid arrangement form a monoid. The associated monoid algebra -- the face algebra -- is well-studied, especially in relation to card shuffling and other Markov chains. In this paper, we explore the action of the symmetric…

Let $G$ be a reductive algebraic group with Lie algebra $\mathfrak{g}$ and $V$ a finite-dimensional representation of $G$. Costello-Gaiotto studied a graded Lie algebra $\mathfrak{d}_{\mathfrak{g}, V}$ and the associated affine Kac-Moody…

In this paper, we define a generalization of Khovanov-Lauda-Rouquier algebras which we call weighted Khovanov-Lauda-Rouquier algebras. We show that these algebras carry many of the same structures as the original Khovanov-Lauda-Rouquier…

We give an overview of the representation theory of restricted rational Cherednik algebras. These are certain finite-dimensional quotients of rational Cherednik algebras at t=0. Their representation theory is connected to the geometry of…

In 1979, Kazhdan and Lusztig developed a combinatorial theory associated with Coxeter groups. They defined in particular partitions of the group in left and two-sided cells. In 1983, Lusztig generalized this theory to Hecke algebras of…

We study the monoid algebra ${}_{n}\mathcal{T}_{m}$ of semistandard Young tableaux, which coincides with the Gelfand--Tsetlin semigroup ring $\mathcal{GT}_{n}$ when $m = n$. Among others, we show that this algebra is commutative,…

In this paper, we study the behavior of categorical actions of a Lie algebra $\mathfrak{g}$ under the deformation of their spectra. We give conditions under which the general point of a family of categorical actions of $\mathfrak{g}$ carry…

The Bethe algebras for the Gaudin model act on the multiplicity space of tensor products of irreducible $ \mathfrak{gl}_r $-modules and have simple spectrum over real points. This fact is proved by Mukhin, Tarasov and Varchenko who also…

We construct a family of involutions on the space $\mathfrak{gl}_n'(\mathbb C)$ of $n\times n$ matrices with real eigenvalues interpolating the complex conjugation and the transpose. We deduce from it a stratified homeomorphism between the…

The face monoid described in [M1] acts on the integrable highest weight modules of a symmetrizable Kac-Moody algebra. It has similar structural properties as a reductive algebraic monoid whose unit group is a Kac-Moody group. We found in…

In this survey article we review Kac-Moody and Heisenberg algebra actions on the categories $\mathcal{O}$ of the rational Cherednik algebras associated to groups $G(\ell,1,n)$. Using these actions we solve basic representation theoretic…

We study the representation theory of the rook-Brauer algebra RB_k(x), also called the partial Brauer algebra. This algebra has a basis of "rook-Brauer" diagrams, which are Brauer diagrams that allow for the possibility of missing edges.…

Elements of a global operator approach to the WZWN theory for compact Riemann surfaces of arbitrary genus $g$ are given. Sheaves of representations of affine Krichever-Novikov algebras over a dense open subset of the moduli space of Riemann…

Quiver varieties have recently appeared in various different areas of Mathematics such as representation theory of Kac-Moody algebras and quantum groups, instantons on 4-manifolds, and resolutions Kleinian singularities. In this paper, we…

We study the action of Bogoliubov transformations on admissible generalized one-particle density matrices arising in Hartree-Fock-Bogoliubov theory. We show that the orbits of this action are reductive homogeneous spaces, and we give…