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)˚.

Maximal Abelian subalgebras of complex Euclidean Lie algebras

1994 ◽  
Vol 72 (7-8) ◽  
pp. 389-404 ◽  
Author(s):  
E. G. Kalnins ◽  
P. Winternitz
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Lie Group ◽  
Conjugacy Classes ◽  
Abelian Subalgebras ◽  
Affine Lie Algebra ◽  
Orthogonal Lie Algebra ◽  
Nilpotent Subalgebras

Maximal Abelian subalgebras (MASAs) of the complex Euclidean Lie algebra [Formula: see text] are classified into conjugacy classes under the action of the Lie group [Formula: see text] Use is made of an earlier classification of MASAs of the orthogonal Lie algebra [Formula: see text] These are then extended to nonsplitting MASAs of [Formula: see text] in which maximal Abelian nilpotent subalgebras of [Formula: see text] are coupled with translations in a nontrivial manner. The methods presented are applicable to the classification of MASAs of any affine Lie algebra.

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]).

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].

scholarly journals Some remarks on invariant Poisson quasi-Nijenhuis structures on Lie groups

2019 ◽  
Vol 16 (07) ◽  
pp. 1950097
Author(s):  
Ghorbanali Haghighatdoost ◽  
Zohreh Ravanpak ◽  
Adel Rezaei-Aghdam
Keyword(s):  
Lie Groups ◽  
Lie Algebras ◽  
Lie Group ◽  
Baxter Equation ◽  
Complex Structures ◽  
Lie Bialgebra ◽  
Text Structures ◽  
Generalized Complex Structures ◽  
Generalized Complex

We study right-invariant (respectively, left-invariant) Poisson quasi-Nijenhuis structures on a Lie group [Formula: see text] and introduce their infinitesimal counterpart, the so-called r-qn structures on the corresponding Lie algebra [Formula: see text]. We investigate the procedure of the classification of such structures on the Lie algebras and then for clarity of our results we classify, up to a natural equivalence, all [Formula: see text]-[Formula: see text] structures on two types of four-dimensional real Lie algebras. We mention some remarks on the relation between [Formula: see text]-[Formula: see text] structures and the generalized complex structures on the Lie algebras [Formula: see text] and also the solutions of modified Yang–Baxter equation (MYBE) on the double of Lie bialgebra [Formula: see text]. The results are applied to some relevant examples.

scholarly journals Classification of left invariant metrics on 4-dimensional solvable Lie groups

2020 ◽  
pp. 14-14
Author(s):  
Tijana Sukilovic
Keyword(s):  
Lie Groups ◽  
Lie Algebras ◽  
Lie Group ◽  
Vector Fields ◽  
Classification Problem ◽  
Inner Product ◽  
Invariant Metrics ◽  
Solvable Lie Groups ◽  
Left Invariant Metrics

In this paper the complete classification of left invariant metrics of arbitrary signature on solvable Lie groups is given. By identifying the Lie algebra with the algebra of left invariant vector fields on the corresponding Lie group ??, the inner product ??,?? on g = Lie G extends uniquely to a left invariant metric ?? on the Lie group. Therefore, the classification problem is reduced to the problem of classification of pairs (g, ??,??) known as the metric Lie algebras. Although two metric algebras may be isometric even if the corresponding Lie algebras are non-isomorphic, this paper will show that in the 4-dimensional solvable case isometric means isomorphic. Finally, the curvature properties of the obtained metric algebras are considered and, as a corollary, the classification of flat, locally symmetric, Ricciflat, Ricci-parallel and Einstein metrics is also given.

Introduction

2012 ◽  
Author(s):  
Mark Green ◽  
Phillip Griffiths ◽  
Matt Kerr
Keyword(s):  
Lie Group ◽  
Hodge Structure ◽  
Algebraic Groups ◽  
Complex Multiplication ◽  
Symmetry Groups ◽  
Hodge Structures ◽  
Fundamental Symmetry ◽  
Variation Of Hodge Structure ◽  
Geometric Origin

This book deals with Mumford-Tate groups, the fundamental symmetry groups in Hodge theory. Much, if not most, of the use of Mumford-Tate groups has been in the study of polarized Hodge structures of level one and those constructed from this case. In this book, Mumford-Tate groups M will be reductive algebraic groups over ℚ such that the derived or adjoint subgroup of the associated real Lie group M ℝ contains a compact maximal torus. In order to keep the statements of the results as simple as possible, the book emphasizes the case when M ℝ itself is semi-simple. The discussion covers period domains and Mumford-Tate domains, the Mumford-Tate group of a variation of Hodge structure, Hodge representations and Hodge domains, Hodge structures with complex multiplication, arithmetic aspects of Mumford-Tate domains, classification of Mumford-Tate subdomains, and arithmetic of period maps of geometric origin.

scholarly journals A note on the conjugacy of Cartan subalgebras

1979 ◽  
Vol 27 (2) ◽  
pp. 163-166
Author(s):  
David J. Winter
Keyword(s):  
Automorphism Group ◽  
Lie Algebra ◽  
Lie Algebras ◽  
Algebraic Groups ◽  
Closed Field ◽  
Algebraically Closed Field ◽  
Cartan Subalgebras ◽  
Cartan Subgroups

AbstractThe conjugacy of Cartan subalgebras of a Lie algebra L over an algebraically closed field under the connected automorphism group G of L is inherited by those G-stable ideals B for which B/Ci is restrictable for some hypercenter Ci of B. Concequently, if L is a restrictable Lie algebra such that L/Ci restrictable for some hypercenter Ci of L, and if the Lie algebra of Aut L contains ad L, then the Cartan subalgebras of L are conjugate under G. (The techniques here apply in particular to Lie algebras of characteristic 0 and classical Lie algebras, showing how the conjugacy of Cartan subgroups of algebraic groups leads quickly in these cases to the conjugacy of Cartan subalgebras.)

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