Conjugacy Classes in Lie Algebras and Algebraic Groups

1967 ◽  
Vol 86 (1) ◽  
pp. 1 ◽  
Author(s):  
R. W. Richardson
Keyword(s):  
Lie Algebras ◽  
Algebraic Groups ◽  
Conjugacy Classes

scholarly journals ON RELATIVE COMPLETE REDUCIBILITY

2020 ◽  
Vol 71 (1) ◽  
pp. 321-334 ◽  
Author(s):  
Christopher Attenborough ◽  
Michael Bate ◽  
Maike Gruchot ◽  
Alastair Litterick ◽  
Gerhard Röhrle
Keyword(s):  
Lie Algebras ◽  
Algebraic Groups ◽  
Reductive Group ◽  
Natural Generalization ◽  
Conjugacy Classes ◽  
Closed Field ◽  
Complete Reducibility ◽  
Algebraic Description ◽  
Reductive Subgroup

Abstract Let $K$ be a reductive subgroup of a reductive group $G$ over an algebraically closed field $k$. The notion of relative complete reducibility, introduced in [M. Bate, B. Martin, G. Röhrle, R. Tange, Complete reducibility and conjugacy classes of tuples in algebraic groups and Lie algebras, Math. Z.269 (2011), no. 1, 809–832], gives a purely algebraic description of the closed $K$-orbits in $G^n$, where $K$ acts by simultaneous conjugation on $n$-tuples of elements from $G$. This extends work of Richardson and is also a natural generalization of Serre’s notion of $G$-complete reducibility. In this paper we revisit this idea, giving a characterization of relative $G$-complete reducibility, which directly generalizes equivalent formulations of $G$-complete reducibility. If the ambient group $G$ is a general linear group, this characterization yields representation-theoretic criteria. Along the way, we extend and generalize several results from [M. Bate, B. Martin, G. Röhrle, R. Tange, Complete reducibility and conjugacy classes of tuples in algebraic groups and Lie algebras, Math. Z.269 (2011), no. 1, 809–832].

Conjugacy Classes of Maximal Tori in Simple Real Algebraic Groups and Applications

1994 ◽  
Vol 46 (4) ◽  
pp. 699-717 ◽  
Author(s):  
Dragomir Ž. Doković ◽  
Nguyêñ Quôć Thăńg
Keyword(s):  
Lie Algebras ◽  
Lie Group ◽  
Short Proof ◽  
Algebraic Groups ◽  
Conjugacy Classes ◽  
Exponential Map ◽  
Simple Complex ◽  
Maximal Tori ◽  
Cartan Subalgebras

AbstractLet G be an almost simple complex algebraic group defined over R, and let G(R) be the group of real points of G. We enumerate the G(R)-conjugacy classes of maximal R-tori of G. Each of these conjugacy classes is also a single G(R)˚-conjugacy class, where G(R)˚ is the identity component of G(R), viewed as a real Lie group. As a consequence we also obtain a new and short proof of the Kostant-Sugiura's theorem on conjugacy classes of Cartan subalgebras in simple real Lie algebras.A connected real Lie group P is said to be weakly exponential (w.e.) if the image of its exponential map is dense in P. This concept was introduced in [HM] where also the question of identifying all w.e. almost simple real Lie groups was raised. By using a theorem of A. Borel and our classification of maximal R-tori we answer the above question when P is of the form G(R)˚.

scholarly journals Complete reducibility and conjugacy classes of tuples in algebraic groups and Lie algebras

2010 ◽  
Vol 269 (3-4) ◽  
pp. 809-832 ◽  
Author(s):  
Michael Bate ◽  
Benjamin Martin ◽  
Gerhard Röhrle ◽  
Rudolf Tange
Keyword(s):  
Lie Algebras ◽  
Algebraic Groups ◽  
Conjugacy Classes ◽  
Complete Reducibility

scholarly journals Reality properties of conjugacy classes in algebraic groups

2008 ◽  
Vol 165 (1) ◽  
pp. 1-27 ◽  
Author(s):  
Anupam Singh ◽  
Maneesh Thakur
Keyword(s):  
Algebraic Groups ◽  
Conjugacy Classes

Quasi-Hereditary Endomorphism Algebras

1995 ◽  
Vol 38 (4) ◽  
pp. 421-428 ◽  
Author(s):  
V. Dlab ◽  
P. Heath ◽  
F. Marko
Keyword(s):  
Lie Algebras ◽  
Algebraic Groups ◽  
Simple Complex ◽  
Highest Weight ◽  
Endomorphism Algebras ◽  
Hereditary Algebras ◽  
Highest Weight Categories

AbstractQuasi-hereditary algebras were introduced by Cline-Parshall-Scott (see [CPS] or [PS]) to deal with highest weight categories which occur in the study of semi-simple complex Lie algebras and algebraic groups. In fact, the quasi-hereditary algebras which appear in these applications enjoy a number of additional properties. The objective of this brief note is to describe a class of lean quasi-hereditary algebras [ADL] which possess such typical characteristics. A study of these questions originated in collaboration with C. M. Ringel (see [DR]).

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 Elements of minimal breadth in finite p-groups and lie algebras

2006 ◽  
Vol 81 (2) ◽  
pp. 209-214 ◽  
Author(s):  
Avinoam Mann
Keyword(s):  
Lie Algebras ◽  
Conjugacy Classes ◽  
Minimal Size ◽  
Nilpotent Lie Algebras

AbstractLet G be a finite p-group, and let M(G) be the subgroup generated by the non-central conjugacy classes of G of minimal size. We prove that this subgroup has class at most 3. A similar result is noted for nilpotent Lie algebras.

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