Block representation type of reduced enveloping algebras

No abstract available

On symmetric invariants of centralisers in reductive Lie algebras

Let $g_e$ be the centraliser of a nilpotent element $e$ in a finite dimensional simple Lie algebra $g$ of rank $l$ over an algebraically closed field of characteristic 0. We investigate the algebra $S(g_e)^{g_e}$ of symmetric invariants of $g_e$ and prove that if $g$ is of type $A$ or $C$, then $S(g_e)^{g_e}$ is always a graded polynomial algebra in $l$ variables. We show that this continues to hold for some nilpotent elements in the Lie algebras of other types. In type $A$ we prove that $S(g_e)^{g_e}$ is freely generated by a regular sequence in $S(g_e)$ and describe the tangent cone at $e$ to the nilpotent variety of $g$.Comment: 49 pages, 2 figure

Non-rationality of some fibrations associated to Klein surfaces

We study the polynomial fibration induced by the equation of the Klein surfaces obtained as quotient of finite linear groups of automorphisms of the plane; this surfaces are of type A, D, E, corresponding to their singularities. The generic fibre of the polynomial fibration is a surface defined over the function field of the line. We proved that it is not rational in cases D, E, although it is obviously rational in the case A. The group of automorphisms of the Klein surfaces is also described, and is linear and of finite dimension in cases D, E; this result being obviously false in case A.Comment: 18 page

The congruence kernel of an arithmetic lattice in a rank one algebraic group over a local field

Let k be a global field and let k_v be the completion of k with respect to v, a non-archimedean place of k. Let \mathbf{G} be a connected, simply-connected algebraic group over k, which is absolutely almost simple of k_v-rank 1. Let G=\mathbf{G}(k_v). Let \Gamma be an arithmetic lattice in G and let C=C(\Gamma) be its congruence kernel. Lubotzky has shown that C is infinite, confirming an earlier conjecture of Serre. Here we provide complete solution of the congruence subgroup problem for \Gamm$ by determining the structure of C. It is shown that C is a free profinite product, one of whose factors is \hat{F}_{\omega}, the free profinite group on countably many generators. The most surprising conclusion from our results is that the structure of C depends only on the characteristic of k. The structure of C is already known for a number of special cases. Perhaps the most important of these is the (non-uniform) example \Gamma=SL_2(\mathcal{O}(S)), where \mathcal{O}(S) is the ring of S-integers in k, with S=\{v\}, which plays a central role in the theory of Drinfeld modules. The proof makes use of a decomposition theorem of Lubotzky, arising from the action of \Gamma on the Bruhat-Tits tree associated with G.Comment: 27 pages, 5 figures, to appear in J. Reine Angew. Mat

Representations of restricted Lie algebras and families of associative ℒ-algebras

Let ℒ be an n-dimensional restricted Lie algebra over an algebraically closed field K of characteristic p > 0. Given a linear function ξ on ℒ and a scalar λ ∈ K, we introduce an associative algebra Uξ,λ(ℒ) of dimension pn over K. The algebra Uξ,1(ℒ) is isomorphic to the reduced enveloping algebra Uξ(ℒ), while the algebra Uξ,0(ℒ) is nothing but the reduced symmetric algebra Sξ(ℒ). Deformation arguments (applied to this family of algebras) enable us to derive a number of results on dimensions of simple ℒ-modules. In particular, we give a new proof of the Kac-Weisfeiler conjecture (see [41], [35]) which uses neither support varieties nor the classification of nilpotent orbits, and compute the maximal dimension of simple ℒ-modules for all ℒ having a toral stabiliser of a linear function

On classical finite and affine W-algebras

This paper is meant to be a short review and summary of recent results on the structure of finite and affine classical W-algebras, and the application of the latter to the theory of generalized Drinfeld-Sokolov hierarchies.Comment: 12 page

Generators of simple Lie algebras in arbitrary characteristics

In this paper we study the minimal number of generators for simple Lie algebras in characteristic 0 or p > 3. We show that any such algebra can be generated by 2 elements. We also examine the 'one and a half generation' property, i.e. when every non-zero element can be completed to a generating pair. We show that classical simple algebras have this property, and that the only simple Cartan type algebras of type W which have this property are the Zassenhaus algebras.Comment: 26 pages, final version, to appear in Math. Z. Main improvements and corrections in Section 4.

Modular Lie algebras and the Gelfand-Kirillov conjecture

Let g be a finite dimensional simple Lie algebra over an algebraically closed field of characteristic zero. We show that if the Gelfand-Kirillov conjecture holds for g, then g has type A_n, C_n or G_2.Comment: 20 page

Laguerre polynomials of derivations

We introduce a grading switching for arbitrary non-associative algebras of prime characteristic p, aimed at producing a new grading of an algebra from a given one. We take inspiration from a fundamental tool in the classification theory of modular Lie algebras known as toral switching, which relies on a delicate adaptation of the exponential of a derivation. Our grading switching is achieved by evaluating certain generalized Laguerre polynomials of degree p − 1, which play the role of generalized exponentials, on a derivation of the algebra. A crucial part of our argument is establishing a congruence for them which is an appropriate analogue of the functional equation exp(x) · exp(y) = exp(x+y) for the classical exponential. Besides having a wider scope, our treatment provides a more transparent explanation of some aspects of the original toral switching, which can be recovered as a special case

Quotients for sheets of conjugacy classes

We provide a description of the orbit space of a sheet S for the conjugation action of a complex simple simply connected algebraic group G. This is obtained by means of a bijection between S 15G and the quotient of a shifted torus modulo the action of a subgroup of the Weyl group and it is the group analogue of a result due to Borho and Kraft. We also describe the normalisation of the categorical quotient // for arbitrary simple G and give a necessary and sufficient condition for //G to be normal in analogy to results of Borho, Kraft and Richardson. The example of G2 is worked out in detail