Spectrum of group cohomology and support varieties

2013 ◽  
Vol 11 (3) ◽  
pp. 507-516 ◽  
Author(s):  
Eric M. Friedlander
Keyword(s):  
Lie Algebras ◽  
Finite Groups ◽  
Equivariant Cohomology ◽  
Group Cohomology ◽  
Qualitative Description ◽  
Spectrum Of Group ◽  
Representations Of Finite Groups ◽  
Restricted Lie Algebras ◽  
Arbitrary Finite Group ◽  
Support Varieties

AbstractWe give a brief introduction to two fundamental papers by Daniel Quillen appearing in the Annals, 1971. These papers established the foundations of equivariant cohomology and gave a qualitative description of the cohomology of an arbitrary finite group. We briefly describe some of the influence of these seminal papers in the study of cohomology and representations of finite groups, restricted Lie algebras, and related structures.

SCHEMES OF TORI AND THE STRUCTURE OF TAME RESTRICTED LIE ALGEBRAS

2001 ◽  
Vol 63 (3) ◽  
pp. 553-570 ◽  
Author(s):  
ROLF FARNSTEINER ◽  
DETLEF VOIGT
Keyword(s):  
Representation Theory ◽  
Lie Algebras ◽  
Classical Case ◽  
Forthcoming Paper ◽  
Structure Theory ◽  
Group Schemes ◽  
Restricted Lie Algebra ◽  
Restricted Lie Algebras ◽  
Geometric Techniques ◽  
Support Varieties

Much of the recent progress in the representation theory of infinitesimal group schemes rests on the application of algebro-geometric techniques related to the notion of cohomological support varieties (cf. [6, 8–10]). The noncohomological characterization of these varieties via the so-called rank varieties (see [21, 22]) involves schemes of additive subgroups that are the infinitesimal counterparts of the elementary abelian groups. In this note we introduce another geometric tool by considering schemes of tori of restricted Lie algebras. Our interest in these derives from the study of infinitesimal groups of tame representation type, whose determination [12] necessitates the results to be presented in §4 and §5 as well as techniques from abstract representation theory.In contrast to the classical case of complex Lie algebras, the information on the structure of a restricted Lie algebra that can be extracted from its root systems is highly sensitive to the choice of the underlying maximal torus. Schemes of tori obviate this defect by allowing us to study algebraic families of root spaces. Accordingly, these schemes may also shed new light on various aspects of the structure theory of restricted Lie algebras. We intend to pursue these questions in a forthcoming paper [13], and focus here on first applications within representation theory.

Finite groups and restricted Lie algebras

1941 ◽  
Vol 8 (3) ◽  
pp. 533-540 ◽  
Author(s):  
Robert Hooke
Keyword(s):  
Lie Algebras ◽  
Finite Groups ◽  
Restricted Lie Algebras

Support varieties for restricted lie algebras

1986 ◽  
Vol 86 (3) ◽  
pp. 553-562 ◽  
Author(s):  
Eric M. Friedlander ◽  
Brian J. Parshall
Keyword(s):  
Lie Algebras ◽  
Restricted Lie Algebras ◽  
Support Varieties

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

1999 ◽  
Vol 1999 (507) ◽  
pp. 189-218 ◽  
Author(s):  
Alexander Premet ◽  
Serge Skryabin
Keyword(s):  
Linear Function ◽  
Lie Algebras ◽  
Associative Algebra ◽  
Maximal Dimension ◽  
Closed Field ◽  
Enveloping Algebra ◽  
Restricted Lie Algebra ◽  
Restricted Lie Algebras ◽  
Support Varieties

Abstract 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.

A Note on Geometric Factoriality

1995 ◽  
Vol 38 (4) ◽  
pp. 390-395 ◽  
Author(s):  
S. M. Bhatwadekar ◽  
K. P. Russell
Keyword(s):  
Finite Groups ◽  
Positive Answer ◽  
Integral Representations ◽  
Perfect Field ◽  
Polynomial Algebras ◽  
Representations Of Finite Groups ◽  
Smooth Affine

AbstractLet k: be a perfect field such that is solvable over k. We show that a smooth, affine, factorial surface birationally dominated by affine 2-space is geometrically factorial and hence isomorphic to . The result is useful in the study of subalgebras of polynomial algebras. The condition of solvability would be unnecessary if a question we pose on integral representations of finite groups has a positive answer.

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