Related papers: Strata of Weyl groups

We define global and local Weyl modules for Lie superalgebras of the form $\mathfrak{g} \otimes A$, where $A$ is an associative commutative unital $\mathbb{C}$-algebra and $\mathfrak{g}$ is a basic Lie superalgebra or $\mathfrak{sl}(n,n)$,…

Let R be the set of isomorphism classes of ideals in the Weyl algebra $A=A_{1}$, and let C be the set of isomorphism classes of triples (V; X, Y), where V is a finite-dimensional (complex) vector space, and X, Y are endomorphisms of V such…

Let $G$ be a connected reductive group over an algebraically closed field with Weyl group $W$. The analogy between Lusztig varieties and Deligne-Lusztig varieties associated to minimal length elements in elliptic conjugacy classes of $W$…

We define Weyl functors, global modules for equivariant map Lie superalgebras $(\g \otimes A)^{\Gamma}$, where $\g$ is basic classical $\mathbb{C}$- Lie superalgebra and $A$ is an associative commutative unital $\mathbb{C}$-algebra. Under…

We classify closed abelian subgroups of the automorphism group of any compact classical simple Lie algebra whose centralizer has the same dimension as the dimension of the subgroup, and describe Weyl groups of maximal abelian subgroups.

A finite W-algebra is an associative algebra constructed from a semisimple Lie algebra and its nilpotent element. In this survey we review recent developments in the representation theory of W-algebras. We emphasize various interactions…

The reflections in a Coxeter group are defined as conjugates of a single generator, and thus admit palindromic expressions as products of generators. Our main result gives closed formulas providing a palindromic reduced expression for each…

The Divisibility Graph of a finite group $G$ has vertex set the set of conjugacy class lengths of non-central elements in $G$ and two vertices are connected by an edge if one divides the other. We determine the connected components of the…

We describe the multiplicative invariant algebras of the root lattices of all irreducible root systems under the action of the Weyl group. In each case, a finite system of fundamental invariants is determined and the class group of the…

We study three fundamental topics in the representation theory of disconnected algebraic groups whose identity component is reductive: (i) the classification of irreducible representations; (ii) the existence and properties of Weyl and dual…

A Lie superalgebrea of Riemannian type leads to a representation of a quadratic Lie algebra into a Weyl algebra. A necessary and sufficient condition that such a representation leads to a Lie superalgebra of Riemannian type is that the…

The orbits of Weyl groups W(B(n)), W(C(n)) and W(D(n)) of the simple Lie algebras B(n), C(n) and D(n) are reduced to the union of the orbits of Weyl groups of the maximal reductive subalgebras of B(n), C(n) and D(n). Matrices transforming…

Let G be a connected, reductive algebraic group over an algebraically closed field of characteristic zero or good and odd. We characterize the spherical conjugacy classes of G as those intersecting only Bruhat cells corresponding to…

By studying the action of the Weyl group of a simple Lie algebra on its root lattice, we construct torsion free subgroups of small and explicitly determined index in a large infinite class of Coxeter groups. One spin-off is the construction…

In a paper by the authors, the associative and the Lie algebras of Weyl type $A[D]=A\otimes F[D]$ were introduced, where $A$ is a commutative associative algebra with an identity element over a field $F$ of any characteristic, and $F[D]$ is…

Suppose $G$ is a reductive algebraic group, $T$ is a Cartan subgroup, $N=\text{Norm}(T)$, and $W=N/T$ is the Weyl group. If $w\in W$ has order $d$, it is natural to ask about the orders lifts of $w$ to $N$. It is straightforward to see that…

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…

We characterize mutation-finite cluster algebras of rank at least 3 using positive semi-definite quadratic forms. In particular, we associate with every unpunctured bordered surface a positive semi-definite quadratic space $V$, and with…

We develop some tools, of an algebraic and combinatorial nature, which enable us to obtain a detailed description of certain quadratic subgroups of the (outer) reduced Weyl group of the Cuntz algebra ${\mathcal O}_n$. In particular, for…

We establish a relation between the known parametrization of a family of irreducible representations of a Weyl group and Springer's correspondence. We outline a parametrization of unipotent character sheaves on a connected reductive group…