Related papers: Minuscule weights and Chevalley groups

The Z-grading determined by a long simple root of an affine or finite type Lie algebra arises from an adjoint or cominuscule representation of a lower rank semi-simple complex Lie algebra. Analysis of the relationship between the grading…

In the paper we prove that every automorphism of and elementary adjoint Chevalley group of types $A_l, D_l$, or $E_l$, over local commutative ring with 1/2 is a composition of a ring automorphism and conjugation by some matrix from the…

Call a compact, connected, simple Lie group $G$ {\emph{adjoint simple}} if it has trivial center. Let $C\subset G$ be a nontrivial conjugacy class, $e\in G$ the identity element of $G$. We prove the existence of an $N\in\mathbb{N}$,…

By algebraic group theory, there is a map from the semisimple conjugacy classes of a finite group of Lie type to the conjugacy classes of the Weyl group. Picking a semisimple class uniformly at random yields a probability measure on…

Let $G(\mathbb{Q})$ be a simply connected Chevalley group over $\mathbb{Q}$ corresponding to a simple Lie algebra $\mathfrak g$ over $\mathbb{C}$. Let $V$ be a finite dimensional faithful highest weight $\mathfrak g$-module and let…

This work is the first in a series of papers devoted to constructing tables of structure constants for the complex simple Lie algebras and to finding an explicit form of Chevalley commutator formulas. The work consists of three parts. In…

Let G be a simple algebraic group over the complex numbers. Let N be the cone of nilpotent elements in the Lie algebra of G. Let K_{G x C^*}(N) denote the Grothendieck group of the category of G x C^*-equivariant coherent sheaves on N. In…

We decompose the adjoint representation of $\mathfrak{sl}_{r+1}=\mathfrak {sl}_{r+1}(\mathbb C)$ by a purely combinatorial approach based on the introduction of a certain subset of the Weyl group called the \emph{Weyl alternation set}…

We classify all simple bounded highest weight modules of a basic classical Lie superalgebra $\mathfrak g$. In particular, our classification leads to the classification of the simple weight modules with finite weight multiplicities over all…

Even though weight multiplicity formulas, such as Kostant's formula, exist their computational use is extremely cumbersome. In fact, even in cases when the multiplicity is well understood, the number of terms considered in Kostant's formula…

In the current article we study structure of a Chevalley group $G(R)$ over a commutative ring $R$. We generalize and improve the following results: (1) standard, relative, and multi-relative commutator formulas; (2) nilpotent structure of…

Let L be the Lie algebra of a simple algebraic group defined over a field F and let H be a split Cartan subalgebra of L. Then L has a Chevalley basis with respect to H. If the characteristic of F is not 2 or 3, it is known how to find it.…

We modify the Hochschild $\phi$-map to construct central extensions of a restricted Lie algebra. Such central extension gives rise to a group scheme which leads to a geometric construction of unrestricted representations. For a classical…

In this paper, we give an explicit formula of Chevalley type, in terms of the Bruhat graph, for the quantum multiplication with the class of the line bundle associated to the anti-dominant minuscule fundamental weight $- \varpi_{k}$ in the…

Let $\mathfrak g$ be a classical Lie superalgebra of type I or a Cartan-type Lie superalgebra {\bf W}$(n)$. We study weight $\mathfrak g$-modules using a method inspired by Mathieu's classification of the simple weight modules with finite…

The goal of this note is to classify the weakly closed unipotent subgroups in the split Chevalley groups. In an application we show under some mild assumptions on the characteristic that the Lie algebra of a connected simple algebraic group…

A Chevalley type integral basis for the ortho-symplectic Lie superalgebra is constructed. The simple modules of the ortho-symplectic supergroup over an algebraically closed field of prime characteristic not equal to 2 are classified, where…

The canonical basis for quantized universal enveloping algebras associated to the finite--dimensional simple Lie algebras, was introduced by Lusztig. The principal technique is the explicit construction (via the braid group action) of a…

As a first application of the double affine Hecke algebra with unequal parameters on Weyl orbits to representation theory of semisimple Lie algebras, we find the graded multiplicities of the trivial module and of the little adjoint module…

We formulate and prove relative versions of several classical decompositions known in the theory of Chevalley groups over commutative rings. As an application we obtain upper estimates for the width of principal congruence subgroups in…