Parabolic subgroup orbits on finite root systems

Affine.m—Mathematica package for computations in representation theory of finite-dimensional and affine Lie algebras

A finite group W generated by reflections is called a finite reflection group and have been studied deeply in different important contexts such as geometry and group theory, matrix algebra and Lie Theory. In this paper we study a class of fuzzy subgroups of finite reflection groups which are called parabolic fuzzy subgroups using preferential equality. In the classical case parabolic subgroups are associated with a set of generators, each a simple reflection, indexed by elements of subsets of a fixed root system Δ . We establish a one-to-one correspondence between the class of fuzzy parabolic subgroups of W and the class of fuzzy subsets of S where S is a set of simple reflections associated with Δ .

Methods and results from the representation theory of finite di- mensional algebras have led to many interactions with other areas of mathe- matics. Such areas include the theory of Lie algebras and quantum groups, commutative algebra, algebraic geometry and topology, and in particular the new theory of cluster algebras. The aim of this workshop was to further de- velop such interactions and to stimulate progress in the representation theory of algebras.

In this talk we introduce generalised Calogero–Moser models and demonstrate their integrability by constructing universal Lax pair operators. These include models based on non-crystallographic root systems, that is the root systems of the finite reflection groups, H 3, H 4, and the dihedral group I 2(m), besides the well-known ones based on crystallographic root systems, namely those associated with Lie algebras. Universal Lax pair operators for all of the generalised Calogero–Moser models and for any choices of the potentials are linear combinations of the reflection operators. The equivalence of the Lax pair with the equations of motion is proved by decomposing the root system into a sum of two-dimensional sub-root systems, A 2, B 2, G 2, and I 2(m). The root type and the minimal type Lax pairs, derived in our previous papers, are given as the simplest representations. The spectral parameter dependence plays an important role in the Lax pair operators, which bear a strong resemblance to the Dunkl operators.

Methods and results from the representation theory of quivers and finite dimensional algebras have led to many interactions with other areas of mathematics. Such areas include the theory of Lie algebras and quantum groups, commutative algebra, algebraic geometry and topology, and in particular the new theory of cluster algebras. The aim of this workshop was to further develop such interactions and to stimulate progress in the representation theory of algebras.

1.1. Let G be a connected reductive algebraic group defined over a finite (prime) field Fp of p elements. For a given rational G-module M one can consider the restrictions of M to the restricted Lie algebra Lie(G) and to the finite group G(Fp) of Fp-rational points. Even though there is no direct functorial connection between the categories of restricted Lie(G)-modules and G(Fp)-modules, the ambient algebraic group G can serve as a bridge linking these two important module categories. Early results of Curtis in the 1960s provided a one-to-one correspondence between the simple restricted modules (over k = Fp) of Lie(G) and those of G(Fp). This correspondence is given by simply restricting certain classes of simple G-modules. More importantly, these results, along with work of Steinberg, allow one to easily transfer the questions pertaining to the computation of irreducible characters among these three categories. For over thirty years, this approach has been one of the main ideas used to relate the representation and cohomology theory of algebraic groups, Lie algebras, and finite groups of Lie type (in the defining characteristic) (see, e.g., [CPSvan], [Jan2], [Hum3], and [Jan1]). Given the previous results in this area, a natural course of investigation should be to find relationships between canonical modules, like projective modules, for Lie(G) and G(Fp). More precisely,

We introduce and study a family of simplicial complexes associated to an arbitrary finite root system and a nonnegative integer parameter m. For m=1, our construction specializes to the (simplicial) generalized associahedra or, equivalently, to the cluster complexes for the cluster algebras of finite type. Our computation of the face numbers and h-vectors of these complexes produces the enumerative invariants defined in other contexts by C.A.Athanasiadis, suggesting links to a host of well studied problems in algebraic combinatorics of finite Coxeter groups, root systems, and hyperplane arrangements. Recurrences satisfied by the face numbers of our complexes lead to combinatorial algorithms for determining Coxeter-theoretic invariants. That is, starting with a Coxeter diagram of a finite Coxeter group, one can compute the Coxeter number, the exponents, and other classical invariants by a recursive procedure that only uses most basic graph-theoretic concepts applied to the input diagram. In types A and B, we rediscover the constructions and results obtained by E.Tzanaki .

As is well known, the Shapovalov bilinear form and its determinant is an important tool in the representation theory of semisimple Lie algebras over char. 0. To our knowledge, the corresponding study of the Shapovalov bilinear form and its determinant is not available in the literature in char. p or the quantum case at roots of unity. The aim of this paper is to fully determine the Shapovalov determinant for both, the restricted enveloping algebra and its quantum analog. More precisely, let g be a semisimple Lie algebra. Fix a prime p6 which also satises p6 whenever g contains a component of type G2. This will be our tacit assumption on p through the paper. Let be a primitive p th root of unity. This paper is concerned with two algebras: a certain analog up of the restricted enveloping algebra (cf. Denition 3:1) and its quantized version u which is an algebra over the cyclotomic eld Q (cf. Denition 3:3). The main results of this paper are complete descriptions of the Shapovalov determinant for both the algebrasup andu (cf. Theorems 3:2 and 3:4).

The basic structure and representation theory of finite-dimensional simple Lie algebras over \(\mathbb{C} \), as developed by W. Killing, E. Cartan and H. Weyl, is a well-known and frequently used branch of mathematics. It is a striking fact that very significant new results in this classical area of mathematics have come to light during the past decade. These new results have originated from the development of the theory of quantum groups. The purpose of this article is to survey briefly the classical results on the representation theory of simple Lie algebras and then to describe the recent developments primarily due to G. Lusztig, M. Kashiwara and P. Littelmann.

Quasi-hereditary algebras were introduced by L. Scott [S] in order to study highest weight categories arising in the representation theory of semisimple complex Lie algebras and algebraic groups, and important results were proved by Cline, Parshall and Scott (see [CPS1,2]). These algebras can be defined entirely in ring-theoretic terms; and they were studied from this point of view by Dlab and Ringel (see [DR1,2], [R1,2]) and by others. In particular it turns out that quasi-hereditary algebras are quite common.

Lie algebras graded by finite irreducible reduced root systems have been classified up to central extensions by Berman and Moody, Benkart and Zelmanov, and Neher. In this paper we determine the central extensions of these Lie algebras and hence describe them completely up to isomorphism.

Various methods of representation theory of modular Lie algebras are improved. As an application the structure of the Lie algebras having a faithful irreducible module of dimension > p 2 > {p^2} is determined. Applications to the classification theory of modular simple Lie algebras are given.

This book presents an introduction to the structure and representation theory of modular Lie algebras over fields of positive characteristic. It introduces the beginner to the theory of modular Lie algebras and is meant to be a reference text for researchers.

Variants of some of the Brauer-Fowler theorems

Reflectable bases for affine reflection systems