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.

Invariant Polynomials of Weyl Groups and Applications to the Centres of Universal Enveloping Algebras

1974 ◽  
Vol 26 (3) ◽  
pp. 583-592 ◽  
Author(s):  
C. Y. Lee
Keyword(s):  
Lie Algebra ◽  
Weyl Groups ◽  
Killing Form ◽  
Dimensional Representation ◽  
Universal Enveloping Algebras ◽  
Semisimple Lie Algebra ◽  
Finite Dimensional Representation ◽  
Polynomial Invariants ◽  
Finite Dimensional ◽  
Complete Set

An element in the centre of the universal enveloping algebra of a semisimple Lie algebra was first constructed by Casimir by means of the Killing form. By Schur's lemma, in an irreducible finite-dimensional representation elements in the centre are represented by diagonal matrices of all whose eigenvalues are equal. In section 2 of this paper, we show the existence of a complete set of generators whose eigenvalues in an irreducible representation are closely related to polynomial invariants of the Weyl group W of the Lie algebra (Theorem 1).

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.

scholarly journals On a chain of reproducing kernel Cartan subalgebras

2021 ◽  
Vol 7 (1) ◽  
pp. 43-49
Author(s):  
Anoh Yannick Kraidi ◽  
Kinvi Kangni
Keyword(s):  
Lie Algebra ◽  
Cartan Subalgebra ◽  
Reproducing Kernel ◽  
Killing Form ◽  
Semisimple Lie Algebra ◽  
Cartan Subalgebras ◽  
A Chain

AbstractLet 𝔤 be a semisimple Lie algebra, j a Cartan subalgebra of 𝔤, j*, the dual of j, jv the bidual of j and B(., .) the restriction to j of the Killing form of 𝔤. In this work, we will construct a chain of reproducing kernel Cartan subalgebras ordered by inclusion.

On the tensor system of a semisimple Lie algebra

1972 ◽  
Vol 71 (2) ◽  
pp. 211-226 ◽  
Author(s):  
Timothy Murphy
Keyword(s):  
Lie Algebra ◽  
Point Of View ◽  
Killing Form ◽  
Essential Characteristic ◽  
Semisimple Lie Algebra ◽  
The Subject

AbstractFrom the tensorial point of view, the essential characteristic of a semisimple lie algebra A is the non-singularity of the Killing form, since this enables one to construct a system of Cartesian tensors over A. That system is the subject of this paper.

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.

A Frobenius homomorphism for Lusztig’s quantum groups for arbitrary roots of unity

2016 ◽  
Vol 18 (03) ◽  
pp. 1550040 ◽  
Author(s):  
Simon Lentner
Keyword(s):  
Lie Algebra ◽  
Roots Of Unity ◽  
Semisimple Lie Algebra ◽  
Braided Category ◽  
Infinite Dimensional ◽  
Enveloping Algebra ◽  
Root Of Unity ◽  
Finite Dimensional ◽  
Nichols Algebras ◽  
Divided Powers

For a finite-dimensional semisimple Lie algebra and a root of unity, Lusztig defined an infinite-dimensional quantum group of divided powers. Under certain restrictions on the order of the root of unity, he constructed a Frobenius homomorphism with finite-dimensional Hopf kernel and with the image of the universal enveloping algebra. In this article, we define and completely describe the Frobenius homomorphism for arbitrary roots of unity by systematically using the theory of Nichols algebras. In several new exceptional cases, the Frobenius–Lusztig kernel is associated to a different Lie algebra than the initial Lie algebra. Moreover, the Frobenius homomorphism often switches short and long roots and/or maps to a braided category.

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.

scholarly journals Polynomial ring representations of endomorphisms of exterior powers

2021 ◽  
Author(s):  
Ommolbanin Behzad ◽  
André Contiero ◽  
Letterio Gatto ◽  
Renato Vidal Martins
Keyword(s):  
Lie Algebra ◽  
Polynomial Ring ◽  
Vertex Operators ◽  
Exterior Power ◽  
Infinite Dimensional ◽  
Symmetric Structure ◽  
Rational Polynomials ◽  
Exterior Powers ◽  
Vertex Representation ◽  
Countably Infinite

AbstractAn explicit description of the ring of the rational polynomials in r indeterminates as a representation of the Lie algebra of the endomorphisms of the k-th exterior power of a countably infinite-dimensional vector space is given. Our description is based on results by Laksov and Throup concerning the symmetric structure of the exterior power of a polynomial ring. Our results are based on approximate versions of the vertex operators occurring in the celebrated bosonic vertex representation, due to Date, Jimbo, Kashiwara and Miwa, of the Lie algebra of all matrices of infinite size, whose entries are all zero but finitely many.

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