Irreducible Representations of Algebras

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.

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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 ◽

Universal Enveloping Algebra ◽

Irreducible Representations ◽

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.

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SQUARES OF WHITE NOISE, SL(2, ℂ) AND KUBO–MARTIN–SCHWINGER STATES

Infinite Dimensional Analysis Quantum Probability and Related Topics ◽

10.1142/s0219025706002597 ◽

2006 ◽

Vol 09 (04) ◽

pp. 491-511 ◽

Cited By ~ 1

Author(s):

DMITRY V. PROKHORENKO

Keyword(s):

White Noise ◽

Universal Enveloping Algebra ◽

Half Line ◽

Probability Measures ◽

Kms States ◽

The Real ◽

Enveloping Algebra ◽

One To One

We investigate the structure of Kubo–Martin–Schwinger (KMS) states on some extension of the universal enveloping algebra of SL (2, ℂ). We find that there exists a one-to-one correspondence between the set of all covariant KMS states on this algebra and the set of all probability measures dμ on the real half-line [0, +∞), which decrease faster than any inverse polynomial. This problem is connected to the problem of KMS states on square of white noise algebra.

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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 ◽

Universal Enveloping Algebra ◽

Irreducible Representations ◽

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.

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PBW Property for Associative Universal Enveloping Algebras Over an Operad

International Mathematics Research Notices ◽

10.1093/imrn/rnaa215 ◽

2020 ◽

Author(s):

Anton Khoroshkin

Keyword(s):

Associative Algebra ◽

Hilbert Series ◽

Abelian Category ◽

Universal Enveloping Algebra ◽

Necessary Condition ◽

Grobner Basis ◽

Universal Enveloping Algebras ◽

Enveloping Algebras ◽

Enveloping Algebra ◽

Category Of Modules

Abstract Given a symmetric operad $\mathcal{P}$ and a $\mathcal{P}$-algebra $V$, the associative universal enveloping algebra ${\textsf{U}_{\mathcal{P}}}$ is an associative algebra whose category of modules is isomorphic to the abelian category of $V$-modules. We study the notion of PBW property for universal enveloping algebras over an operad. In case $\mathcal{P}$ is Koszul a criterion for the PBW property is found. A necessary condition on the Hilbert series for $\mathcal{P}$ is discovered. Moreover, given any symmetric operad $\mathcal{P}$, together with a Gröbner basis $G$, a condition is given in terms of the structure of the underlying trees associated with leading monomials of $G$, sufficient for the PBW property to hold. Examples are provided.

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An Irreducible Representation of sl(2)

Canadian Mathematical Bulletin ◽

10.4153/cmb-1974-010-x ◽

1974 ◽

Vol 17 (1) ◽

pp. 63-65

Author(s):

F.W. Lemire

Keyword(s):

Irreducible Representation ◽

Zero Element ◽

Universal Enveloping Algebra ◽

Irreducible Representations ◽

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.

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CONSTRUCTIVE NONCOMMMUTATIVE INVARIANT THEORY

Transformation Groups ◽

10.1007/s00031-021-09643-2 ◽

2021 ◽

Author(s):

MÁTYÁS DOMOKOS ◽

VESSELIN DRENSKY

Keyword(s):

Lie Algebra ◽

Associative Algebra ◽

Invariant Theory ◽

Reductive Group ◽

Symmetric Tensor ◽

Free Associative Algebra ◽

Tensor Algebra ◽

Enveloping Algebra ◽

Finite Dimensional ◽

Degree Bounds

AbstractThe problem of finding generators of the subalgebra of invariants under the action of a group of automorphisms of a finite-dimensional Lie algebra on its universal enveloping algebra is reduced to finding homogeneous generators of the same group acting on the symmetric tensor algebra of the Lie algebra. This process is applied to prove a constructive Hilbert–Nagata Theorem (including degree bounds) for the algebra of invariants in a Lie nilpotent relatively free associative algebra endowed with an action induced by a representation of a reductive group.

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