scholarly journals Bilinear and Quadratic Forms on Rational Modules of Split Reductive Groups

2016 ◽  
Vol 68 (2) ◽  
pp. 395-421 ◽  
Author(s):  
Skip Garibaldi ◽  
Daniel K. Nakano
Keyword(s):  
Lie Groups ◽  
Lie Algebras ◽  
Quadratic Forms ◽  
Reductive Groups ◽  
Compact Lie Groups ◽  
Tilting Modules ◽  
Weyl Modules ◽  
Semisimple Complex ◽  
Characteristic 2 ◽  
Semisimple Algebraic Groups

AbstractThe representation theory of semisimple algebraic groups over the complex numbers (equivalently, semisimple complex Lie algebras or Lie groups, or real compact Lie groups) and the questions of whether a given complex representation is symplectic or orthogonal have been solved since at least the 1950s. Similar results for Weyl modules of split reductive groups over fields of characteristic different from z hold by using similar proofs. This paper considers analogues of these results for simple, induced, and tilting modules of split reductive groups over fields of prime characteristic as well as a complete answer for Weyl modules over fields of characteristic 2.

Introduction

2015 ◽  
Author(s):  
Brian Conrad ◽  
Gopal Prasad
Keyword(s):  
Quadratic Forms ◽  
General Structure ◽  
Root Systems ◽  
Simple Groups ◽  
Reductive Groups ◽  
Central Extensions ◽  
New Techniques ◽  
Characteristic 2 ◽  
Minimal Type

This book deals with the classification of pseudo-reductive groups. Using new techniques and constructions, it addresses a number of questions; for example, whether there are versions of the Isomorphism and Isogeny Theorems for pseudosplit pseudo-reductive groups and of the Existence Theorem for pseudosplit pseudo-simple groups; whether the automorphism functor of a pseudo-semisimple group is representable; or whether there is a Tits-style classification in the pseudo-semisimple case recovering the version due to Tits in the semisimple case. This introduction discusses the special challenges of characteristic 2 as well as root systems, exotic groups and degenerate quadratic forms, and tame central extensions. It also reviews generalized standard groups, minimal type and general structure theorem, and Galois-twisted forms and Tits classification.

scholarly journals POINCARÉ NORMAL FORMS AND SIMPLE COMPACT LIE GROUPS

2002 ◽  
Vol 17 (25) ◽  
pp. 3571-3587 ◽  
Author(s):  
GIUSEPPE GAETA
Keyword(s):  
Dynamical Systems ◽  
Lie Groups ◽  
Lie Algebras ◽  
Normal Forms ◽  
Linear Part ◽  
Compact Lie Groups ◽  
Fundamental Representation ◽  
Normalizing Transformations

We classify the possible behavior of Poincaré–Dulac normal forms for dynamical systems in Rn with nonvanishing linear part and which are equivariant under (the fundamental representation of) all the simple compact Lie algebras and thus the corresponding simple compact Lie groups. The "renormalized forms" (in the sense of Ref. 22) of these systems is also discussed; in this way we are able to simplify the classification and moreover to analyze systems with zero linear part. We also briefly discuss the convergence of the normalizing transformations.

scholarly journals Some Questions about Semisimple Lie Groups Originating in Matrix Theory

2003 ◽  
Vol 46 (3) ◽  
pp. 332-343 ◽  
Author(s):  
Dragomir Ž. Ðoković ◽  
Tin-Yau Tam
Keyword(s):  
Partial Order ◽  
Lie Groups ◽  
Lie Algebras ◽  
Matrix Theory ◽  
Compact Subgroup ◽  
Orbit Space ◽  
Natural Partial Order ◽  
Semisimple Lie Groups ◽  
Complex Matrix ◽  
Semisimple Complex

AbstractWe generalize the well-known result that a square traceless complex matrix is unitarily similar to a matrix with zero diagonal to arbitrary connected semisimple complex Lie groups G and their Lie algebras under the action of a maximal compact subgroup K of G. We also introduce a natural partial order on : x ≤ y if f (K · x) ⊆ f (K · y) for all f 2 *, the complex dual of . This partial order is K-invariant and induces a partial order on the orbit space /K. We prove that, under some restrictions on , the set f (K · x) is star-shaped with respect to the origin.

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