On the Structure of a Class of Graded Modules Related to Symmetric Pairs

2006 ◽  
Vol 13 (02) ◽  
pp. 315-328 ◽  
Author(s):  
Gang Han
Keyword(s):  
Lie Algebra ◽  
Adjoint Representation ◽  
Exterior Algebra ◽  
Symmetric Algebra ◽  
Module Structure ◽  
Cartan Decomposition ◽  
Isotropy Representation ◽  
Semisimple Lie Algebra ◽  
Graded Modules ◽  
Special Case

Let [Formula: see text] be the Cartan decomposition of a real semisimple Lie algebra and [Formula: see text] be its complexification. Let [Formula: see text] be the corresponding isotropy representation, and the exterior algebra [Formula: see text] becomes a graded [Formula: see text]-module by extending ν. We study a graded [Formula: see text]-submodule C of [Formula: see text] and get two important decompositions of the [Formula: see text]-module [Formula: see text]. Let [Formula: see text] be the symmetric algebra over [Formula: see text]. Then [Formula: see text] also has an [Formula: see text]-module structure, which is [Formula: see text]-equivariant, and C is a space of generators for this module. Our results generalize Kostant's results in the special case that ν is the adjoint representation of a semisimple Lie algebra.

ISOTROPY REPRESENTATIONS, EIGENVALUES OF A CASIMIR ELEMENT, AND COMMUTATIVE LIE SUBALGEBRAS

2001 ◽  
Vol 64 (1) ◽  
pp. 61-80 ◽  
Author(s):  
DMITRI I. PANYUSHEV
Keyword(s):  
Lie Algebra ◽  
Finite Order ◽  
Killing Form ◽  
Maximal Eigenvalue ◽  
Isotropy Representation ◽  
Semisimple Lie Algebra ◽  
Casimir Element ◽  
Root Of Unity ◽  
Exterior Powers

Let [hfr ] be a reductive subalgebra of a semisimple Lie algebra [gfr ] and C[hfr ] ∈ U([hfr ]) be the Casimir element determined by the restriction of the Killing form on [gfr ] to [hfr ]. The paper studies eigenvalues of C[hfr ] on the isotropy representation [mfr ]≃[gfr ]/[hfr ]. Some general estimates connecting the eigenvalues and the Dynkin indices of [mfr ] are given. If [hfr ] is a symmetric subalgebra, it is shown that describing the maximal eigenvalue of C[hfr ] on exterior powers of [mfr ] is connected with possible dimensions of commutative Lie subalgebras in [mfr ], thereby extending a result of Kostant. In this situation, a formula is also given for the maximal eigenvalue of C[hfr ] on ∧ [mfr ]. More generally, a similar picture arises if [hfr ] = [gfr ]Θ, where Θ is an automorphism of finite order m and [mfr ] is replaced by the eigenspace of Θ corresponding to a primitive mth root of unity.

On a Conjecture of Goresky, Kottwitz and MacPherson

1999 ◽  
Vol 51 (1) ◽  
pp. 3-9 ◽  
Author(s):  
C. Allday ◽  
V. Puppe
Keyword(s):  
Exterior Algebra ◽  
Symmetric Algebra ◽  
Koszul Duality ◽  
Graded Modules

AbstractWe settle a conjecture of Goresky, Kottwitz and MacPherson related to Koszul duality, i.e., to the correspondence between differential graded modules over the exterior algebra and those over the symmetric algebra.

scholarly journals The adjoint representation inside the exterior algebra of a simple Lie algebra

2015 ◽  
Vol 280 ◽  
pp. 21-46 ◽  
Author(s):  
Corrado De Concini ◽  
Paolo Papi ◽  
Claudio Procesi
Keyword(s):  
Lie Algebra ◽  
Adjoint Representation ◽  
Exterior Algebra ◽  
Simple Lie Algebra

Exterior Powers of the Adjoint Representation

1997 ◽  
Vol 49 (1) ◽  
pp. 133-159 ◽  
Author(s):  
Mark Reeder
Keyword(s):  
Lie Algebra ◽  
Adjoint Representation ◽  
Irreducible Representations ◽  
Semisimple Lie Algebra ◽  
Complex Semisimple ◽  
Exterior Powers

AbstractExterior powers of the adjoint representation of a complex semisimple Lie algebra are decomposed into irreducible representations, to varying degrees of satisfaction.

ANTI-SELF-DUAL SOLUTIONS OF THE YANG-MILLS EQUATIONS IN 4n DIMENSIONS

1992 ◽  
Vol 07 (23) ◽  
pp. 2077-2085 ◽  
Author(s):  
A. D. POPOV
Keyword(s):  
Scalar Field ◽  
Euclidean Space ◽  
Lie Algebra ◽  
Linear Equations ◽  
Dual Solutions ◽  
Gauge Fields ◽  
System Of Equations ◽  
Semisimple Lie Algebra ◽  
Yang Mills ◽  
Self Duality

The anti-self-duality equations for gauge fields in d = 4 and a generalization of these equations to dimension d = 4n are considered. For gauge fields with values in an arbitrary semisimple Lie algebra [Formula: see text] we introduce the ansatz which reduces the anti-self-duality equations in the Euclidean space ℝ4n to a system of equations breaking up into the well known Nahm's equations and some linear equations for scalar field φ.

PBW deformations of quantum symmetric algebras and their group extensions

2016 ◽  
Vol 15 (03) ◽  
pp. 1650049 ◽  
Author(s):  
Piyush Shroff ◽  
Sarah Witherspoon
Keyword(s):  
Finite Group ◽  
Lie Algebra ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Group Extensions ◽  
Enveloping Algebra ◽  
Symmetric Algebras ◽  
Necessary And Sufficient ◽  
Special Case

We examine PBW deformations of finite group extensions of quantum symmetric algebras, in particular the quantum Drinfeld orbifold algebras defined by the first author. We give a homological interpretation, in terms of Gerstenhaber brackets, of the necessary and sufficient conditions on parameter functions to define a quantum Drinfeld orbifold algebra, thus clarifying the conditions. In case the acting group is trivial, we determine conditions under which such a PBW deformation is a generalized enveloping algebra of a color Lie algebra; our PBW deformations include these algebras as a special case.

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