DECOMPOSITION OF SYMMETRIC AND EXTERIOR POWERS OF THE ADJOINT REPRESENTATION OF ${\mathfrak g}l_N$ 1: UNIMODALITY OF PRINCIPAL SPECIALIZATION OF THE INTERNAL PRODUCT OF THE SCHUR FUNCTIONS

1992 ◽  
Vol 07 (supp01b) ◽  
pp. 545-579 ◽  
Author(s):  
ANATOL N. KIRILLOV
Keyword(s):  
Lie Algebra ◽  
Schur Functions ◽  
Exact Formula ◽  
Adjoint Representation ◽  
Combinatorial Approach ◽  
Rigged Configurations ◽  
Irreducible Components ◽  
Exterior Powers ◽  
Principal Specialization

The problem of decomposing the symmetric and exterior algebras of the adjoint representation of the Lie algebra [Formula: see text] into [Formula: see text]-irreducible components are considered. The exact formula for the principal specialization of the internal product of the Schur functions (similar to the formula for Kostka-Foulkes polynomials) is obtained by the purely combinatorial approach, based on the theory of rigged configurations. The stable behaviour of some polynomials is studied. Different examples are presented.

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.

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.

scholarly journals Rigged configurations and cylindric loop Schur functions

2018 ◽  
Vol 5 (4) ◽  
pp. 513-555 ◽  
Author(s):  
Thomas Lam ◽  
Pavlo Pylyavskyy ◽  
Reiho Sakamoto
Keyword(s):  
Schur Functions ◽  
Rigged Configurations

scholarly journals Representations of Bihom-Lie Algebras

2022 ◽  
Vol 29 (01) ◽  
pp. 125-142
Author(s):  
Yongsheng Cheng ◽  
Huange Qi
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Adjoint Representation ◽  
Trivial Representation ◽  
Central Extensions ◽  
Linear Maps

A Bihom-Lie algebra is a generalized Hom-Lie algebra endowed with two commuting multiplicative linear maps. In this paper, we study representations of Bihom-Lie algebras. In particular, derivations, central extensions, derivation extensions, the trivial representation and the adjoint representation of Bihom-Lie algebras are studied in detail.

scholarly journals Yangians versus minimal W-algebras: A surprising coincidence

2020 ◽  
pp. 2050036
Author(s):  
Victor G. Kac ◽  
Pierluigi Möseneder Frajria ◽  
Paolo Papi
Keyword(s):  
Lie Algebra ◽  
Monic Polynomial ◽  
Adjoint Representation ◽  
Conformal Weight ◽  
Quantum Fields ◽  
Simple Lie Algebra ◽  
Finite Dimensional ◽  
Matrix Formula ◽  
Polynomial Formula

We prove that the singularities of the [Formula: see text]-matrix [Formula: see text] of the minimal quantization of the adjoint representation of the Yangian [Formula: see text] of a finite dimensional simple Lie algebra [Formula: see text] are the opposite of the roots of the monic polynomial [Formula: see text] entering in the OPE expansions of quantum fields of conformal weight [Formula: see text] of the universal minimal affine [Formula: see text]-algebra at level [Formula: see text] attached to [Formula: see text].

The eightfold way model, the SU(3)-flavor model and the medium–strong interaction

2015 ◽  
Vol 30 (12) ◽  
pp. 1550050
Author(s):  
Syed Afsar Abbas
Keyword(s):  
Quantum Chromodynamics ◽  
Lie Algebra ◽  
Strong Interaction ◽  
Baryon Number ◽  
Adjoint Representation ◽  
Macroscopic Models ◽  
Physical Representation ◽  
New Perspective ◽  
The Way

Lack of any baryon number in the eightfold way model, and its intrinsic presence in the SU(3)-flavor model, has been a puzzle since the genesis of these models in 1961–1964. First we show that the conventional popular understanding of this puzzle is actually fundamentally wrong, and hence the problem being so old, begs urgently for resolution. In this paper we show that the issue is linked to the way that the adjoint representation is defined mathematically for a Lie algebra, and how it manifests itself as a physical representation. This forces us to distinguish between the global and the local charges and between the microscopic and the macroscopic models. As a bonus, a consistent understanding of the hitherto mysterious medium–strong interaction is achieved. We also gain a new perspective on how confinement arises in quantum chromodynamics.

MATHEMATICA™ PACKAGES FOR COMPUTING PRINCIPAL DECOMPOSITIONS OF SIMPLE LIE ALGEBRAS AND APPLICATIONS IN EXTENDED CONFORMAL FIELD THEORIES

2003 ◽  
Vol 14 (01) ◽  
pp. 1-27 ◽  
Author(s):  
DANIELA GĂRĂJEU ◽  
MIHAIL GĂRĂJEU
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Three Dimensional ◽  
Adjoint Representation ◽  
Cartan Matrix ◽  
Field Theories ◽  
Conformal Field Theories ◽  
Killing Form ◽  
Simple Lie Algebras ◽  
Conformal Field

In this article, we propose two Mathematica™ packages for doing calculations in the domain of classical simple Lie algebras. The main goal of the first package, [Formula: see text], is to determine the principal three-dimensional subalgebra of a simple Lie algebra. The package provides several functions which give some elements related to simple Lie algebras (generators in fundamental and adjoint representation, roots, Killing form, Cartan matrix, etc.). The second package, [Formula: see text], concerns the principal decomposition of a Lie algebra with respect to the principal three-dimensional embedding. These packages have important applications in extended two-dimensional conformal field theories. As an example, we present an application in the context of the theory of W-gravity.

scholarly journals Lie Invariants in Two and Three Variables

2014 ◽  
Vol 21 (01) ◽  
pp. 95-116
Author(s):  
Murray R. Bremner ◽  
Jiaxiong Hu
Keyword(s):  
Lie Algebra ◽  
Computer Algebra ◽  
Coefficient Matrix ◽  
Adjoint Representation ◽  
Free Lie Algebra ◽  
Natural Representation ◽  
3 Dimensional

We use computer algebra to determine the Lie invariants of degree ≤ 12 in the free Lie algebra on two generators corresponding to the natural representation of the simple 3-dimensional Lie algebra 𝔰𝔩2(ℂ). We then consider the free Lie algebra on three generators, and compute the Lie invariants of degree ≤ 7 corresponding to the adjoint representation of 𝔰𝔩2(ℂ), and the Lie invariants of degree ≤ 9 corresponding to the natural representation of 𝔰𝔩3(ℂ). We represent the action of 𝔰𝔩2(ℂ) and 𝔰𝔩3(ℂ) on Lie polynomials by computing the coefficient matrix with respect to the basis of Hall words.

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