Related papers: Affine $G_2$ Centralizer Algebras

Let S be a principally embedded sl_2 subalgebra in sl_n for n > 2. A special case of results of the third author and Gregg Zuckerman implies that there exists a positive integer b(n) such that for any finite-dimensional irreducible sl_n…

An approach to representations of finite groups is presented without recourse to character theory. Considering the group algebra C[G] as an algebra of linear maps on C[G] (by left multiplication), we derive the primitive central idempotents…

For positive integers $u$ and $v$, let $L_u=\begin{bmatrix} 1 & 0 \\ u & 1 \end{bmatrix}$ and $R_v=\begin{bmatrix} 1 & v \\ 0 & 1 \end{bmatrix}$. Let $S_{u,v}$ be the monoid generated by $L_u$ and $R_v$, and $G_{u,v}$ be the group generated…

Two Lie algebroids are presented that are linked to the construction of the linearizing output of an affine in the input nonlinear system. The algorithmic construction of the linearizing output proceeds inductively, and each stage has two…

The group $G_2$ of invertible affine transformations of $\mathbb{R}^2$ has, up to equivalence, one square--integrable representation. Two new realizations of this representation are presented and novel continuous wavelet transforms, acting…

It is well-known that affine Hecke algebras are very useful to describe the smooth representations of any connected reductive p-adic group G, in terms of the supercuspidal representations of its Levi subgroups. The goal of this paper is to…

In this paper we study a family of commutative algebras generated by two infinite sets of generators. These algebras are parametrized by Young diagrams. We explain a connection of these algebras with the fusion product of integrable…

For each simple Lie algebra g consider the corresponding affine vertex algebra V_{crit}(g) at the critical level. The center of this vertex algebra is a commutative associative algebra whose structure was described by a remarkable theorem…

Given a group $G$ and an integer $n\geq2$ we construct a new group $\tilde{{\cal K}}(G,n)$. Although this construction naturally occurs in the context of finding new invariants for complex algebraic surfaces, it is related to the theory of…

Each irreducible representation of the affine group of a finite field has a unique maximal inductive algebra, and it is self adjoint.

Let $A$ be a finite dimensional unital commutative associative algebra and let $B$ be a finite dimensional vertex $A$-algebroid such that its Levi factor is isomorphic to $sl_2$. Under suitable conditions, we construct an indecomposable…

Continuing earlier work, we show how to realize irreducible finite-dimensional representations of the complex group of type $G_2$ via tableaux, along the way exhibiting explicit generators of the defining ideal of the flag variety

Let $\mathfrak{g}$ be a complex simple Lie algebra and $Z(\mathfrak{g})$ be the center of the universal enveloping algebra $U(\mathfrak{g})$. Denote by $V_\lambda$ the finite-dimensional irreducible $\mathfrak{g}$-module with highest weight…

We construct a new family of affine $W$-algebras $W^k(\lambda,\mu)$ parameterized by partitions $\lambda$ and $\mu$ associated with the centralizers of nilpotent elements in $\mathfrak{gl}_N$. The new family unifies a few known classes of…

Gaudin algebra is the commutative subalgebra in $U(\mathfrak{g})^{\otimes N}$ generated by higher integrals of the quantum Gaudin magnet chain attached to a semisimple Lie algebra $\mathfrak{g}$. This algebra depends on a collection of…

The Grassmannian of affine subspaces is a natural generalization of both the Euclidean space, points being zero-dimensional affine subspaces, and the usual Grassmannian, linear subspaces being special cases of affine subspaces. We show…

We first construct an action of the extended double affine braid group $\mathcal{\ddot{B}}$ on the quantum toroidal algebra $U_{q}(\mathfrak{g}_{\mathrm{tor}})$ in untwisted and twisted types. As a crucial step in the proof, we obtain a…

We construct level one representations of the quantum affine algebra $U_q(G_2^{(1)})$ by vertex operators from bosonic fields.

We introduce the first examples of groups $G$ with infinite center which in a natural sense are completely recognizable from their von Neumann algebras, $\mathcal{L}(G)$. Specifically, assume that $G=A\times W$, where $A$ is an infinite…

In this article we define $G$-algebras, that is, graded algebras on which a reductive group $G$ acts as gradation preserving automorphisms. Starting from a finite dimensional $G$-module $V$ and the polynomial ring $\mathbb{C}[V]$, it is…