The adjoint representation of a classical Lie algebra and the support of Kostant’s weight multiplicity formula

2016 ◽  
Vol 7 (1) ◽  
pp. 75-116
Author(s):  
Pamela E. Harris ◽  
Erik Insko ◽  
Lauren Kelly Williams
Keyword(s):  
Lie Algebra ◽  
Adjoint Representation ◽  
Weight Multiplicity ◽  
Multiplicity Formula ◽  
Classical Lie Algebra

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 When is the $q$-Multiplicity of a Weight a Power of $q$?

10.37236/8758 ◽  
2019 ◽  
Vol 26 (4) ◽  
Author(s):  
Pamela E. Harris ◽  
Margaret Rahmoeller ◽  
Lisa Schneider ◽  
Anthony Simpson
Keyword(s):  
Lie Algebras ◽  
Dynkin Diagram ◽  
Fibonacci Numbers ◽  
Simple Lie Algebras ◽  
Weight Multiplicity ◽  
Highest Weight ◽  
Multiplicity Formula ◽  
Exhaustive List

Berenshtein and Zelevinskii provided an exhaustive list of pairs of weights $(\lambda,\mu)$ of simple Lie algebras $\mathfrak{g}$ (up to Dynkin diagram isomorphism) for which the multiplicity of the weight $\mu$ in the representation of $\mathfrak{g}$ with highest weight $\lambda$ is equal to one. Using Kostant's weight multiplicity formula we describe and enumerate the contributing terms to the multiplicity for subsets of these pairs of weights and show that, in these cases, the cardinality of these contributing sets is enumerated by (multiples of) Fibonacci numbers. We conclude by using these results to compute the associated $q$-multiplicity for the pairs of weights considered, and conjecture that in all cases the $q$-multiplicity of such pairs of weights is given by a power of $q$.

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.

ON ROOT MULTIPLICITIES OF $H A_n^{(1)}$

2002 ◽  
Vol 12 (03) ◽  
pp. 477-508 ◽  
Author(s):  
JENNIFER HONTZ ◽  
KAILASH C. MISRA
Keyword(s):  
Lie Algebra ◽  
Irreducible Representations ◽  
Root Multiplicities ◽  
Multiplicity Formula

We determine the root multiplicities of the Kac–Moody Lie algebra [Formula: see text] of indefinite type using a recursive root multiplicity formula due to Kang. We view [Formula: see text] as a representation of its subalgebra [Formula: see text] and then use the combinatorics of the irreducible representations of [Formula: see text] to determine the root multiplicities.

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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