Related papers: Minuscule weights and Chevalley groups

We investigate Beurling-Fourier algebras, a weighted version of Fourier algebras, on various Lie groups focusing on their spectral analysis. We will introduce a refined general definition of weights on the dual of locally compact groups and…

The goal of this paper is to compute the supports of simple modules in the categories $\mathcal{O}$ for the rational Cherednik algebras associated to groups $G(\ell,1,n)$. For this we compute some combinatorial maps on the set of simples:…

The theory of a two-valued algebraic group structure on a complex plane and complex projective line is developed. In this theory, depending on the choice of the neutral element, the local multiplication law is given by the Buchstaber…

In this paper, we construct, investigate and, in some cases, classify several new classes of (simple) modules over the Takiff $\mathfrak{sl}_{2}$. More precisely, we first explicitly construct and classify, up to isomorphism, all modules…

We introduce a new partial order on the set of all antichains of a fixed size in any poset. When applied to minuscule posets, these partial orders give rise to distributive lattices that appear in the branching rules for minuscule…

We present explicit upper bounds for the number and size of conjugacy classes in finite Chevalley groups and their variations. These results have been used by many authors to study zeta functions associated to representations of finite…

The weight multiplicities of finite dimensional simple Lie algebras can be computed individually using various methods. Still, it is hard to derive explicit closed formulas. Similarly, explicit closed formulas for the multiplicities of…

We continue study of subgroups of a Chevalley group $G_P(\Phi,R)$ over a ring $R$ with a root system $\Phi$ and a weight lattice $P$, containing the elementary subgroup $E_P(\Phi,K)$ over a subring $K$ of $R$. Recently A. Bak and A.…

We study the analogue of Sidki's weak commutativity construction, defined originally for groups, in the category of Lie algebras. This is the quotient $\chi(\mathfrak{g})$ of the Lie algebra freely generated by two isomorphic copies…

Color Lie algebras, which were introduced by Ree, are a graded extension of Lie (super)algebras by an abelian group. We show that the color Lie algebras can be used to construct universal weight systems for knot invariants of of Vassiliev…

We show that the different labelings of the crystal graph for irreducible highest weight $\mathcal{U}\_q (\hat{\mathfrak{sl}}\_e)$-modules lead to different labelings of the simple modules for non semisimple Ariki-Koike algebras by using…

We study the action of the Chevalley involution of a simple complex Lie group G on the set of its weight varieties (i.e. torus quotients of its flag manifolds). We find for torus quotients of Grassmannians (for GL(n)) that we obtain the…

Let G be a reductive algebraic group over a field of prime characteristic. One can associate to G (or subgroups thereof) its Lie algebra, its Frobenius kernels, and the finite Chevalley group of points over a finite field. The…

Finite group theorists have established many formulas that express interesting properties of a finite group in terms of sums of characters of the group. An obstacle to applying these formulas is lack of control over the dimensions of…

Using crossed homomorphisms, we show that the category of weak representations (resp. admissible representations) of Lie-Rinehart algebras (resp. Leibniz pairs) is a left module category over the monoidal category of representations of Lie…

Let $G$ be a simple algebraic group over an algebraically closed field $K$ with Lie algebra $\mathfrak{g}$. For unipotent elements $u \in G$ and nilpotent elements $e \in \mathfrak{g}$, the Jordan block sizes of $\operatorname{Ad}(u)$ and…

To any simple Lie algebra $\mathfrak g$ and automorphism $\sigma:\mathfrak g\to \mathfrak g$ we associate a cyclotomic Gaudin algebra. This is a large commutative subalgebra of $U(\mathfrak g)^{\otimes N}$ generated by a hierarchy of…

Let $\mathfrak{g}$ be a nilpotent Lie algebra which is also regarded as a homogeneous Lie group with the Campbell-Hausdorff multiplication. This allows to define a generalized multiplication $f \# g = (f^{\vee} * g^{\vee})^{\wedge}$ of two…

A Lie algebra $L$ is said to be $(\Theta_{n},sl_{n})$-graded if it contains a simple subalgebra $\mathfrak{g}$ isomorphic to $sl_{n}$ such that the $\mathfrak{g}$-module $L$ decomposes into copies of the adjoint module, the trivial module,…

In this paper we prove that every automorphism of a Chevalley group (or its elementary subgroup) with root system of rank >1 over a commutative ring (with 1/2 for the systems A_2, F_4, B_l, C_l; with 1/2 and 1/3 for the system G_2) is…