arXiv subjects

Alexander Premet

Publications and source records attributed to Alexander Premet.

25 records · Page 2Linked to original sources

arXiv2007

Primitive ideals, non-restricted representations and finite W-algebras

We prove that all finite W-algebras associated with nilpotent elements e in a complex semisimple Lie algebra g have finite-dimensional representations. In order to obtain this result we establish a connection between primitive ideals of U(g) attached to the nilpotent orbit containing e and finite-dimensional representations of the reduced enveloping algebra assiciated with e over an algebraically closed field of finite characteristic.

math.RT↗

arXiv2006

Classification of finite dimensional simple Lie algebras in prime characteristics

We give a comprehensive survey of the theory of finite dimensional Lie algebras over an algebraically closed field of characteristic p>0 and announce that for p>3 the classification of finite dimensional simple Lie algebras is complete. Any such Lie algebra is up to isomorphism either classical (i.e. comes from characteristic 0) or a filtered Lie algebra of Cartan type or a Melikian algebra of characteristic 5.

math.RA↗

arXiv2006

Simple Lie algebras of small characteristic V. The non-Melikian case

Let L be a finite dimensional simple Lie algebra over an algebraically closed field of characteristic p>3. We prove in this paper that if all tori of maximal dimansion in the semisimple p-envelope of L are standard, the L is up to isomorphism either a Lie algebra of type A_n, B_n, C_n, D_n, E_6, E_7, E_8, F_4, G_2 or a filtered Lie algebra of Cartan type. When p>5, the assumption on tori is fulfilled automatically and our result can be restated as follows: every finite dimensional simple Lie algebra over an algebraically closee field of characteristic p>5 is either classical or of Cartan type. This theorem settles the last remaining case p=7 of the Kostrikin-Shafarevich conjecture (from 1966).

math.RA↗

arXiv2005

The Recognition Theorem for Graded Lie Algebras of Prime Characteristic

The Recognition Theorem for graded Lie algebras is an essential ingredient in the classification of finite-dimensional simple Lie algebras over an algebraically closed field of characteristic p > 3. The main goal of this monograph is to present the first complete proof of this fundamental result.

math.RA↗

arXiv2005

Enveloping algebras of Slodowy slices and the Joseph ideal

We study general properties of quantisations of Slodowy slices and discuss in detail the case of the minimal nilpotent orbit. Associated varieties of related primitive ideals of U(g) are determined, in the general case, and the de Vos-van Driel conjecture on Verma modules for finite W-algebras is proved in the minimal nilpotent case.

math.RT↗

arXiv2004

Zassenhaus varieties of general linear Lie algebras

Let g be a Lie algebra over an algebraically closed field of characteristic p>0 and let U(g) be the universal enveloping algebra of g. We prove in this paper that for g=gl_n and g=sl_n the centre of U(g) is a unique factorisation domain and its field of fractions is rational. For g=sl_n our argument requires the assumption that p\nmid n while for g=gl_n it works for any p. It turned out that our two main results are closely related to each other. The first one confirms in type ${\rm A}$ a recent conjecture of A.Braun and C.Hajarnavis while the second answers a question of J.Alev.

math.RA↗

arXiv2003

Nilpotent commuting varieties of reductive Lie algebras

We prove that the nilpotent commuting variety of a reductive Lie algebra over an algebraically closed field of good characteristic is equidimensional. In characteristic zero, this confirms a conjecture of Vladimir Baranovsky. As a by-product, we obtain tat the punctual (local) Hilbert scheme parametrising the ideals of colength $n$ in $k[[X,Y]]$ is irreducible over any algebraically closed field $k$.

math.RT↗