An Irreducible Representation of sl(2)

Enveloping Algebra

In a recent paper [1] MM. Arnal and Pinczon have classified all complex irreducible representations (ρ, V) of sl(2) having the property (P) that there exists a non-zero element x∈sl(2) such that ρ(x) admits an eigenvalue. It is the purpose of this note to demonstrate, by example, that there exist irreducible representations of sl(2) which do not have property (P). As usual, we consider sl(2) embedded in its universal enveloping algebra U and identify the representations of sl(2) and U.

TWO PERMANENTS IN THE UNIVERSAL ENVELOPING ALGEBRAS OF THE SYMPLECTIC LIE ALGEBRAS

International Journal of Mathematics ◽

10.1142/s0129167x09005327 ◽

2009 ◽

Vol 20 (03) ◽

pp. 339-368 ◽

Cited By ~ 2

Author(s):

MINORU ITOH

Keyword(s):

Lie Algebra ◽

Lie Algebras ◽

General Linear ◽

Universal Enveloping Algebras ◽

Enveloping Algebras ◽

Enveloping Algebra ◽

Central Elements ◽

Orthogonal Lie Algebra

This paper presents new generators for the center of the universal enveloping algebra of the symplectic Lie algebra. These generators are expressed in terms of the column-permanent and it is easy to calculate their eigenvalues on irreducible representations. We can regard these generators as the counterpart of central elements of the universal enveloping algebra of the orthogonal Lie algebra given in terms of the column-determinant by Wachi. The earliest prototype of all these central elements is the Capelli determinants in the universal enveloping algebra of the general linear Lie algebra.

Irreducible Representations of Algebras

Canadian Journal of Mathematics ◽

10.4153/cjm-1974-104-0 ◽

1974 ◽

Vol 26 (5) ◽

pp. 1118-1129 ◽

Cited By ~ 1

Author(s):

Edgar G. Goodaire

Keyword(s):

Associative Algebra ◽

Enveloping Algebra ◽

Representations Of Algebras ◽

One To One

The concept of the universal enveloping algebra of a (not necessarily associative) algebra X is basic to the study of the representations of X, because there is a one-to-one correspondence between the representations of X and . If one is only interested in studying a certain class of the representations of X, the thought occurs that there may exist a more suitable universal object.

Classification theorem on irreducible representations of theq-deformed algebraU′q(son)

International Journal of Mathematics and Mathematical Sciences ◽

10.1155/ijmms.2005.225 ◽

2005 ◽

Vol 2005 (2) ◽

pp. 225-262 ◽

Cited By ~ 8

Author(s):

N. Z. Iorgov ◽

A. U. Klimyk

Keyword(s):

Lie Algebra ◽

Complete Classification ◽

Classical Type ◽

Enveloping Algebra ◽

Root Of Unity ◽

Complete Reducibility ◽

Finite Dimensional

The aim of this paper is to give a complete classification of irreducible finite-dimensional representations of the nonstandardq-deformationU′q(son)(which does not coincide with the Drinfel'd-Jimbo quantum algebraUq(son)) of the universal enveloping algebraU(son(ℂ))of the Lie algebrason(ℂ)whenqis not a root of unity. These representations are exhausted by irreducible representations of the classical type and of the nonclassical type. The theorem on complete reducibility of finite-dimensional representations ofU′q(son)is proved.

H-Finite Irreducible Representations of Simple Lie Algebras

Canadian Journal of Mathematics ◽

10.4153/cjm-1979-100-5 ◽

1979 ◽

Vol 31 (5) ◽

pp. 1084-1106 ◽

Cited By ~ 2

Author(s):

F. Lemire ◽

M. Pap

Keyword(s):

Irreducible Representation ◽

Lie Algebras ◽

Complex Number ◽

Cartan Subalgebra ◽

Weight Space ◽

Algebra Homomorphism ◽

One Dimensional ◽

Enveloping Algebra ◽

Complex Number Field

Let L denote a simple Lie algebra over the complex number field C with H a fixed Cartan subalgebra and C(L) the centralizer of H in the universal enveloping algebra U of L. It is known [cf. 2, 5] that one can construct from each algebra homomorphism ϕ:C(L) → C a unique algebraically irreducible representation of L which admits a weight space decomposition relative to H in which the weight space corresponding to ϕ ↓ H ∈ H* is one-dimensional. Conversely, if (ρ, V) is an algebraically irreducible representation of L admitting a one-dimensional weight space Vλ for some λ ∈ H*, then there exists a unique algebra homomorphism ϕ:C(L) → C which extends λ such that (ρ, V) is equivalent to the representation constructed from ϕ. Any such representation will be said to be pointed.

A decomposition formula for representations

Nagoya Mathematical Journal ◽

10.1017/s0027763000002543 ◽

1987 ◽

Vol 107 ◽

pp. 63-68 ◽

Cited By ~ 2

Author(s):

George Kempf

Keyword(s):

Irreducible Representation ◽

General Formula ◽

Parabolic Subgroup ◽

Characteristic Zero ◽

Maximal Torus ◽

Reductive Group ◽

Levi Subgroup ◽

Know How ◽

Decomposition Formula

Let H be the Levi subgroup of a parabolic subgroup of a split reductive group G. In characteristic zero, an irreducible representation V of G decomposes when restricted to H into a sum V = ⊕mαWα where the Wα’s are distinct irreducible representations of H. We will give a formula for the multiplicities mα. When H is the maximal torus, this formula is Weyl’s character formula. In theory one may deduce the general formula from Weyl’s result but I do not know how to do this.