The Adjoint Representation of a Lie Group

1993 ◽  
pp. 245-246
Author(s):  
Albert S. Schwarz
Keyword(s):  
Lie Group ◽  
Adjoint Representation

scholarly journals Lie Supergroups Obtained from 3-Dimensional Lie Superalgebras Associated to the Adjoint Representation and Having a 2-Dimensional Derived Ideal

2008 ◽  
Vol 2008 ◽  
pp. 1-9
Author(s):  
I. Hernández ◽  
R. Peniche
Keyword(s):  
Lie Group ◽  
Lie Superalgebra ◽  
Adjoint Representation ◽  
Lie Superalgebras ◽  
Base Manifold ◽  
3 Dimensional ◽  
Lie Supergroups

We give the explicit multiplication law of the Lie supergroups for which the base manifold is a 3-dimensional Lie group and whose underlying Lie superalgebrag=g0⊕g1which satisfiesg1=g0,g0acts ong1via the adjoint representation andg0has a 2-dimensional derived ideal.

scholarly journals Low codimensional embeddings of Sp(n) and Su(n)

1984 ◽  
Vol 27 (1) ◽  
pp. 25-29 ◽  
Author(s):  
G. Walker ◽  
R. M. W. Wood
Keyword(s):  
Euclidean Space ◽  
Lie Group ◽  
Maximal Torus ◽  
Unitary Group ◽  
Normal Bundle ◽  
Symplectic Group ◽  
Adjoint Representation ◽  
General Technique ◽  
Special Cases ◽  
Connected Lie Group

In [4] Elmer Rees proves that the symplectic group Sp(n) can be smoothly embedded in Euclidean space with codimension 3n, and the unitary group U(n) with codimension n. These are special cases of a result he obtains for a compact connected Lie group G. The general technique is first to embed G/T, where T is a maximal torus, as a maximal orbit of the adjoint representation of G, and then to extendto an embedding of G by using a maximal orbit of a faithful representation of G. In thisnote, we observe that in the cases G = Sp(n) or SU(n) an improved result is obtained byusing the “symplectic torus” S3 x … x S3 in place of T = S1 x … x S1. As in Rees's construction, the normal bundle of the embedding of G is trivial.

scholarly journals FIELD OF INVARIANTS OF BORELEAN GROUP OF ADJOINT REPRESENTATION OF GL(n,K)

2017 ◽  
Vol 20 (3) ◽  
pp. 34-40
Author(s):  
K.A. Vyatkina
Keyword(s):  
Explicit Form ◽  
Lie Group ◽  
Invariant Theory ◽  
Algebraic Independence ◽  
Adjoint Representation ◽  
Unitriangular Group ◽  
Joint Representation

The paper is devoted to invariant theory problems, in particular to the problem of finding generators of invariant fields in an explicit form. The set of generators is given for invariant field of unitriangular group concerning the ad-joint representation of GL(n, K) group. Moreover, the set of generators of Borel group for the field of invariants is constructed and their algebraic independence is proved. Lie group;adjoint representation;field of invariant;generators of the field of invariants;Borel group;

A homotopy construction of the adjoint representation for Lie groups

2002 ◽  
Vol 133 (3) ◽  
pp. 399-409 ◽  
Author(s):  
NATÀLIA CASTELLANA ◽  
NITU KITCHLOO
Keyword(s):  
Lie Groups ◽  
Lie Group ◽  
Maximal Torus ◽  
Maximal Rank ◽  
Adjoint Representation ◽  
Simply Connected

Let G be a compact, simply-connected, simple Lie group and T ⊂ G a maximal torus. The purpose of this paper is to study the connection between various fibrations over BG (where G is a compact, simply-connected, simple Lie group) associated to the adjoint representation and homotopy colimits over poset categories [Cscr ], hocolim[Cscr ]BGI where GI are certain connected maximal rank subgroups of G.

scholarly journals Hom-Lie Algebras and Hom-Lie Groups, Integration and Differentiation

2020 ◽  
Author(s):  
Jun Jiang ◽  
◽  
Satyendra Kumar Mishra ◽  
Yunhe Sheng ◽  
◽  
...  
Keyword(s):  
Lie Groups ◽  
Lie Algebra ◽  
Lie Algebras ◽  
Group Action ◽  
Lie Group ◽  
Smooth Manifold ◽  
Adjoint Representation ◽  
Lie Group Action

In this paper, we introduce the notion of a (regular) Hom-Lie group. We associate a Hom-Lie algebra to a Hom-Lie group and show that every regular Hom-Lie algebra is integrable. Then, we define a Hom-exponential (Hexp) map from the Hom-Lie algebra of a Hom-Lie group to the Hom-Lie group and discuss the universality of this Hexp map. We also describe a Hom-Lie group action on a smooth manifold. Subsequently, we give the notion of an adjoint representation of a Hom-Lie group on its Hom-Lie algebra. At last, we integrate the Hom-Lie algebra (gl(V),[.,.],Ad), and the derivation Hom-Lie algebra of a Hom-Lie algebra.

On the Unitarized Adjoint Representation of a Semisimple Lie Group II

1977 ◽  
Vol 29 (6) ◽  
pp. 1217-1222
Author(s):  
Ronald L. Lipsman
Keyword(s):  
Lebesgue Measure ◽  
Lie Algebra ◽  
Unitary Representation ◽  
Lie Group ◽  
Adjoint Action ◽  
Adjoint Representation ◽  
Semisimple Lie Group ◽  
Group Ii ◽  
Image Position

Let G be a connected semisimple Lie group with Lie algebra . Lebesgue measure on is invariant under the adjoint action of G; and so there is a natural unitary representation TG of G on L2 given by

Symmetry properties of fractional order transport equations

2012 ◽  
Vol 9 (1) ◽  
pp. 59-64
Author(s):  
R.K. Gazizov ◽  
A.A. Kasatkin ◽  
S.Yu. Lukashchuk
Keyword(s):  
Differential Equations ◽  
Fractional Differential Equations ◽  
Lie Group ◽  
Initial Conditions ◽  
Group Analysis ◽  
Equivalence Transformation ◽  
Point Change ◽  
Infinitesimal Operator ◽  
Symmetry Properties ◽  
Fractional Differential

In the paper some features of applying Lie group analysis methods to fractional differential equations are considered. The problem related to point change of variables in the fractional differentiation operator is discussed and some general form of transformation that conserves the form of Riemann-Liouville fractional operator is obtained. The prolongation formula for extending an infinitesimal operator of a group to fractional derivative with respect to arbitrary function is presented. Provided simple example illustrates the necessity of considering both local and non-local symmetries for fractional differential equations in particular cases including the initial conditions. The equivalence transformation forms for some fractional differential equations are discussed and results of group classification of the wave-diffusion equation are presented. Some examples of constructing particular exact solutions of fractional transport equation are given, based on the Lie group methods and the method of invariant subspaces.

ε‎-Invariance

2018 ◽  
Author(s):  
Ercüment H. Ortaçgil
Keyword(s):  
Lie Algebra ◽  
Lie Group

The discussions up to Chapter 4 have been concerned with the Lie group. In this chapter, the Lie algebra is constructed by defining the operators ∇ and ∇̃.

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