scholarly journals Geometric representation theory of restricted Lie algebras

2001 ◽  
Vol 6 (2) ◽  
pp. 175-191 ◽  
Author(s):  
I. Mirković ◽  
D. Rumynin
Keyword(s):  
Representation Theory ◽  
Lie Algebras ◽  
Geometric Representation ◽  
Geometric Representation Theory ◽  
Restricted Lie Algebras

Borel subalgebras of restricted Cartan-Type Lie algebras

2021 ◽  
pp. 2250210
Author(s):  
Ke Ou ◽  
Bin Shu
Keyword(s):  
Representation Theory ◽  
Lie Algebras ◽  
Open Problem ◽  
Conjugacy Classes ◽  
Classical Type ◽  
Geometric Representation ◽  
Cartan Type ◽  
Geometric Representation Theory ◽  
Geometric Setting

It is still an open problem to determine the conjugacy classes of Borel subalgebras of non-classical type Lie algebras. In this paper, we prove that there are at least 2 conjugacy classes of Borel subalgebras as well as maximal triangulable subalgebras of restricted Cartan type Lie algebras of type W, S and H. We are particularly interested in maximal triangulable subalgebras of [Formula: see text] under some conditions which is called [Formula: see text]-subalgebras (Definition 3.1). We classify the conjugacy classes of [Formula: see text]-subalgebras for [Formula: see text] and determine their representatives. This paper and its sequel [Z. Lin, K. Ou and B. Shu, Geometric Setting of Jacobson–Witt Algebras, preprint] attempt to establish both algebraic and geometric setting for geometric representation theory of [Formula: see text]

Enveloping Algebras and Geometric Representation Theory

2015 ◽  
Vol 12 (2) ◽  
pp. 1385-1447 ◽  
Author(s):  
Iain Gordon ◽  
Bernard Leclerc ◽  
Wolfgang Soergel
Keyword(s):  
Representation Theory ◽  
Geometric Representation ◽  
Enveloping Algebras ◽  
Geometric Representation Theory

scholarly journals Mixed motives and geometric representation theory in equal characteristic

2019 ◽  
Vol 25 (2) ◽  
Author(s):  
Jens Niklas Eberhardt ◽  
Shane Kelly
Keyword(s):  
Representation Theory ◽  
Geometric Representation ◽  
Geometric Representation Theory ◽  
Mixed Motives

Geometric Representation Theory and Gauge Theory

2019 ◽  
Author(s):  
Alexander Braverman ◽  
Michael Finkelberg ◽  
Andrei Negut ◽  
Alexei Oblomkov
Keyword(s):  
Gauge Theory ◽  
Representation Theory ◽  
Geometric Representation ◽  
Geometric Representation Theory

Enveloping Algebras and Geometric Representation Theory

2012 ◽  
Vol 9 (1) ◽  
pp. 733-809
Author(s):  
Shrawan Kumar ◽  
Peter Littelmann ◽  
Wolfgang Soergel
Keyword(s):  
Representation Theory ◽  
Geometric Representation ◽  
Enveloping Algebras ◽  
Geometric Representation Theory

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.

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