On the second cohomology of an algebraic group and of its lie algebra in a positive characteristic

2017 ◽  
Vol 101 (5-6) ◽  
pp. 841-849
Author(s):  
Sh. Sh. Ibraev
Keyword(s):  
Algebraic Group ◽  
Lie Algebra ◽  
Positive Characteristic

scholarly journals The Square-Zero Basis of Matrix Lie Algebras

Mathematics ◽  
2020 ◽  
Vol 8 (6) ◽  
pp. 1032
Author(s):  
Raúl Durán Díaz ◽  
Víctor Gayoso Martínez ◽  
Luis Hernández Encinas ◽  
Jaime Muñoz Masqué
Keyword(s):  
Algebraic Group ◽  
Lie Algebra ◽  
Lie Algebras ◽  
Positive Characteristic ◽  
Linear Algebraic Group ◽  
Invariant Functions ◽  
Arbitrary Characteristic ◽  
Independent Invariant ◽  
Field Of Positive Characteristic

A method is presented that allows one to compute the maximum number of functionally-independent invariant functions under the action of a linear algebraic group as long as its Lie algebra admits a basis of square-zero matrices even on a field of positive characteristic. The class of such Lie algebras is studied in the framework of the classical Lie algebras of arbitrary characteristic. Some examples and applications are also given.

scholarly journals Varieties of a class of elementary subalgebras

2021 ◽  
Vol 7 (2) ◽  
pp. 2084-2101
Author(s):  
Yang Pan ◽  
◽  
Yanyong Hong ◽  
Keyword(s):  
Algebraic Group ◽  
Lie Algebra ◽  
Positive Characteristic ◽  
Maximal Dimension ◽  
Closed Field ◽  
Characteristic P ◽  
Simple Algebraic Group ◽  
Restricted Lie Algebra ◽  
Irreducible Components ◽  
Type C

<abstract><p>Let $ G $ be a connected standard simple algebraic group of type $ C $ or $ D $ over an algebraically closed field $ \Bbbk $ of positive characteristic $ p &gt; 0 $, and $ \mathfrak{g}: = \mathrm{Lie}(G) $ be the Lie algebra of $ G $. Motivated by the variety of $ \mathbb{E}(r, \mathfrak{g}) $ of $ r $-dimensional elementary subalgebras of a restricted Lie algebra $ \mathfrak{g} $, in this paper we characterize the irreducible components of $ \mathbb{E}(\mathrm{rk}_{p}(\mathfrak{g})-1, \mathfrak{g}) $ where the $ p $-rank $ \mathrm{rk}_{p}(\mathfrak{g}) $ is defined to be the maximal dimension of an elementary subalgebra of $ \mathfrak{g} $.</p></abstract>

scholarly journals RELATIVE COMPLETE REDUCIBILITY AND NORMALIZED SUBGROUPS

2020 ◽  
Vol 8 ◽  
Author(s):  
MAIKE GRUCHOT ◽  
ALASTAIR LITTERICK ◽  
GERHARD RÖHRLE
Keyword(s):  
Automorphism Group ◽  
Algebraic Group ◽  
Lie Algebra ◽  
Reductive Algebraic Group ◽  
Identity Component ◽  
Complete Reducibility ◽  
Closed Fields ◽  
Completely Reducible ◽  
Reductive Subgroup ◽  
The Absolute

We study a relative variant of Serre’s notion of $G$ -complete reducibility for a reductive algebraic group $G$ . We let $K$ be a reductive subgroup of $G$ , and consider subgroups of $G$ that normalize the identity component $K^{\circ }$ . We show that such a subgroup is relatively $G$ -completely reducible with respect to $K$ if and only if its image in the automorphism group of $K^{\circ }$ is completely reducible. This allows us to generalize a number of fundamental results from the absolute to the relative setting. We also derive analogous results for Lie subalgebras of the Lie algebra of $G$ , as well as ‘rational’ versions over nonalgebraically closed fields.

scholarly journals Spherical subgroups in simple algebraic groups

2015 ◽  
Vol 151 (7) ◽  
pp. 1288-1308
Author(s):  
Friedrich Knop ◽  
Gerhard Röhrle
Keyword(s):  
Algebraic Group ◽  
Closed Subgroup ◽  
Positive Characteristic ◽  
Dense Orbit ◽  
Simple Algebraic Group ◽  
Spherical Subgroup ◽  
Group A ◽  
General Deformation ◽  
Characteristic 2

Let $G$ be a simple algebraic group. A closed subgroup $H$ of $G$ is said to be spherical if it has a dense orbit on the flag variety $G/B$ of $G$. Reductive spherical subgroups of simple Lie groups were classified by Krämer in 1979. In 1997, Brundan showed that each example from Krämer’s list also gives rise to a spherical subgroup in the corresponding simple algebraic group in any positive characteristic. Nevertheless, up to now there has been no classification of all such instances in positive characteristic. The goal of this paper is to complete this classification. It turns out that there is only one additional instance (up to isogeny) in characteristic 2 which has no counterpart in Krämer’s classification. As one of our key tools, we prove a general deformation result for subgroup schemes that allows us to deduce the sphericality of subgroups in positive characteristic from the same property for subgroups in characteristic zero.

scholarly journals The scheme of Lie sub-algebras of a Lie algebra and the equivariant cotangent map

1974 ◽  
Vol 53 ◽  
pp. 59-70
Author(s):  
William J. Haboush
Keyword(s):  
Algebraic Group ◽  
Lie Algebra ◽  
Lie Algebras ◽  
Algebraic Variety ◽  
Local Properties

The main object of this paper is to develop techniques for investigating the local properties of actions of an algebraic group on an algebraic variety. Our main tools are certain schemes which may be associated to Lie algebras.

Classes of unipotent elements in simple algebraic groups. I

1976 ◽  
Vol 79 (3) ◽  
pp. 401-425 ◽  
Author(s):  
P. Bala ◽  
R. W. Carter
Keyword(s):  
Algebraic Group ◽  
Lie Algebra ◽  
Algebraic Groups ◽  
Adjoint Action ◽  
Conjugacy Classes ◽  
Closed Field ◽  
Algebraically Closed Field ◽  
Nilpotent Elements

LetGbe a simple adjoint algebraic group over an algebraically closed fieldK. We are concerned to describe the conjugacy classes of unipotent elements ofG. Goperates on its Lie algebra g by means of the adjoint action and we may consider classes of nilpotent elements of g under this action. It has been shown by Springer (11) that there is a bijection between the unipotent elements ofGand the nilpotent elements ofgwhich preserves theG-action, provided that the characteristic ofKis either 0 or a ‘good prime’ forG. Thus we may concentrate on the problem of classifying the nilpotent elements of g under the adjointG-action.

scholarly journals Computing with Nilpotent Orbits in Simple Lie Algebras of Exceptional Type

2008 ◽  
Vol 11 ◽  
pp. 280-297 ◽  
Author(s):  
Willem A. de Graaf
Keyword(s):  
Algebraic Group ◽  
Lie Algebra ◽  
Lie Algebras ◽  
Simple Algorithm ◽  
Adjoint Action ◽  
Closed Field ◽  
Nilpotent Orbits ◽  
Simple Algebraic Group ◽  
Algebraically Closed Field ◽  
Simple Lie Algebras

AbstractLet G be a simple algebraic group over an algebraically closed field with Lie algebra g. Then the orbits of nilpotent elements of g under the adjoint action of G have been classified. We describe a simple algorithm for finding a representative of a nilpotent orbit. We use this to compute lists of representatives of these orbits for the Lie algebras of exceptional type. Then we give two applications. The first one concerns settling a conjecture by Elashvili on the index of centralizers of nilpotent orbits, for the case where the Lie algebra is of exceptional type. The second deals with minimal dimensions of centralizers in centralizers.

scholarly journals Group gradings on the Lie algebra psln in positive characteristic

2009 ◽  
Vol 213 (9) ◽  
pp. 1739-1749 ◽  
Author(s):  
Yuri Bahturin ◽  
Mikhail Kochetov
Keyword(s):  
Lie Algebra ◽  
Positive Characteristic ◽  
Group Gradings
Sign in / Sign up

Export Citation Format

Share Document