scholarly journals Affine.m—Mathematica package for computations in representation theory of finite-dimensional and affine Lie algebras

2012 ◽  
Vol 183 (11) ◽  
pp. 2480-2493 ◽  
Author(s):  
Anton Nazarov
Keyword(s):  
Representation Theory ◽  
Lie Algebras ◽  
Affine Lie Algebras ◽  
Finite Dimensional

Representations of affine Lie algebras and soliton equations

1985 ◽  
Vol 315 (1533) ◽  
pp. 391-391
Keyword(s):  
Representation Theory ◽  
Lie Algebras ◽  
Gordon Equation ◽  
Vries Equation ◽  
Affine Lie Algebras ◽  
Loop Groups ◽  
Sine Gordon Equation ◽  
Kyoto School ◽  
Hidden Symmetries ◽  
Soliton Equations

A few years ago the 'hidden symmetries’ of the soliton equations had been identified as affine Lie groups, also known as loop groups. The first extensive use of the representation theory of affine Lie algebras for the soliton equations have been developed in a series of works by mathematicians of the Kyoto school. We will review some of their results and develop them further on the basis of the representation theory. Thus an orbit of the simplest affine Lie group SL(2, C)^ in the fundamental representation V will provide the solutions of the Korteweg-de Vries equation, and similarly the solutions of the sine-Gordon equation will come from an orbit of the group (SL(2, C) x SL(2, C)) ^ in V x V*.

scholarly journals Coloured Generalised Young Diagrams for Affine Weyl-Coxeter Groups

10.37236/931 ◽  
2007 ◽  
Vol 14 (1) ◽  
Author(s):  
R. C. King ◽  
T. A. Welsh
Keyword(s):  
Lie Algebras ◽  
Coxeter Group ◽  
Coxeter Groups ◽  
Slight Variation ◽  
Affine Lie Algebras ◽  
Young Diagrams ◽  
Bijective Correspondence ◽  
Coxeter Number ◽  
Finite Dimensional ◽  
Dual Coxeter Number

Coloured generalised Young diagrams $T(w)$ are introduced that are in bijective correspondence with the elements $w$ of the Weyl-Coxeter group $W$ of $\mathfrak{g}$, where $\mathfrak{g}$ is any one of the classical affine Lie algebras $\mathfrak{g}=A^{(1)}_\ell$, $B^{(1)}_\ell$, $C^{(1)}_\ell$, $D^{(1)}_\ell$, $A^{(2)}_{2\ell}$, $A^{(2)}_{2\ell-1}$ or $D^{(2)}_{\ell+1}$. These diagrams are coloured by means of periodic coloured grids, one for each $\mathfrak{g}$, which enable $T(w)$ to be constructed from any expression $w=s_{i_1}s_{i_2}\cdots s_{i_t}$ in terms of generators $s_k$ of $W$, and any (reduced) expression for $w$ to be obtained from $T(w)$. The diagram $T(w)$ is especially useful because $w(\Lambda)-\Lambda$ may be readily obtained from $T(w)$ for all $\Lambda$ in the weight space of $\mathfrak{g}$. With $\overline{\mathfrak{g}}$ a certain maximal finite dimensional simple Lie subalgebra of $\mathfrak{g}$, we examine the set $W_s$ of minimal right coset representatives of $\overline{W}$ in $W$, where $\overline{W}$ is the Weyl-Coxeter group of $\overline{\mathfrak{g}}$. For $w\in W_s$, we show that $T(w)$ has the shape of a partition (or a slight variation thereof) whose $r$-core takes a particularly simple form, where $r$ or $r/2$ is the dual Coxeter number of $\mathfrak{g}$. Indeed, it is shown that $W_s$ is in bijection with such partitions.

A New Class of Representations of EALA Coordinated by Quantum Tori in Two Variables

2002 ◽  
Vol 45 (4) ◽  
pp. 672-685 ◽  
Author(s):  
S. Eswara Rao ◽  
Punita Batra
Keyword(s):  
Lie Algebras ◽  
Primitive Root ◽  
Irreducible Module ◽  
Affine Lie Algebras ◽  
Quantum Torus ◽  
New Class ◽  
Root Of Unity ◽  
Quantum Tori ◽  
Finite Dimensional

AbstractWe study the representations of extended affine Lie algebras where q is N-th primitive root of unity (ℂq is the quantum torus in two variables). We first prove that ⊕ for a suitable number of copies is a quotient of . Thus any finite dimensional irreducible module for ⊕ lifts to a representation of . Conversely, we prove that any finite dimensional irreducible module for comes from above. We then construct modules for the extended affine Lie algebras which is integrable and has finite dimensional weight spaces.

scholarly journals Simple Finite-Dimensional Modules and Monomial Bases from the Gelfand-Testlin Patterns

2021 ◽  
pp. 60-65
Author(s):  
Amadou Keita
Keyword(s):  
Representation Theory ◽  
Lie Algebras ◽  
Simple Module ◽  
Explicit Construction ◽  
Research Area ◽  
The Past ◽  
Finite Dimensional ◽  
Dimensional Module

One of the most important classes of Lie algebras is sl_n, which are the n×n matrices with trace 0. The representation theory for sl_n has been an interesting research area for the past hundred years and in it, the simple finite-dimensional modules have become very important. They were classified and Gelfand and Tsetlin actually gave an explicit construction of a basis for every simple finite-dimensional module. This paper extends their work by providing theorems and proofs and constructs monomial bases of the simple module.

scholarly journals GENERAL BRANCHING FUNCTIONS OF AFFINE LIE ALGEBRAS

1995 ◽  
Vol 10 (10) ◽  
pp. 823-830 ◽  
Author(s):  
STEPHEN HWANG ◽  
HENRIC RHEDIN
Keyword(s):  
Lie Algebras ◽  
Simple Proof ◽  
Affine Lie Algebras ◽  
Finite Dimensional ◽  
Brst Cohomology ◽  
Special Case ◽  
Finite Dimensional Algebras ◽  
Explicit Expressions

Explicit expressions are presented for general branching functions for cosets of affine Lie algebras ĝ with respect to subalgebras ĝ′ for the cases where the corresponding finite-dimensional algebras g and g′ are such that g is simple and g′ is either simple or sums of u(1) terms. A special case of the latter yields the string functions. Our derivation is purely algebraical and has its origin in the results on the BRST cohomology presented by us earlier. We will here give an independent and simple proof of the validity of our results. The method presented here generalizes in a straightforward way to more complicated g and g′ such as sums of simple and u(1) terms.

Representations of ω-Lie algebras and tailed derivations of Lie algebras

2020 ◽  
pp. 1-15
Author(s):  
Runxuan Zhang
Keyword(s):  
Representation Theory ◽  
Lie Algebra ◽  
Lie Algebras ◽  
Irreducible Module ◽  
Nonassociative Algebra ◽  
Complex Field ◽  
Indecomposable Modules ◽  
Finite Dimensional ◽  
Nilpotent Matrices ◽  
One To One

We study the representation theory of finite-dimensional [Formula: see text]-Lie algebras over the complex field. We derive an [Formula: see text]-Lie version of the classical Lie’s theorem, i.e., any finite-dimensional irreducible module of a soluble [Formula: see text]-Lie algebra is 1-dimensional (1D). We also prove that indecomposable modules of some 3D [Formula: see text]-Lie algebras could be parametrized by the complex field and nilpotent matrices. We introduce the notion of a tailed derivation of a nonassociative algebra [Formula: see text] and prove that if [Formula: see text] is a Lie algebra, then there exists a one-to-one correspondence between tailed derivations of [Formula: see text] and 1D [Formula: see text]-extensions of [Formula: see text].

scholarly journals Equivalence of certain categories of modules for quantized affine lie algebras

2000 ◽  
Vol 69 (2) ◽  
pp. 162-175
Author(s):  
Vyacheslav M. Futorny ◽  
Duncan J. Melville
Keyword(s):  
Quantum Group ◽  
Lie Algebras ◽  
Verma Module ◽  
Verma Modules ◽  
Affine Lie Algebras ◽  
Finite Part ◽  
Moody Algebra ◽  
Finite Dimensional ◽  
Equivalence Of Categories ◽  
Category Of Modules

AbstractWe show that a quantum Verma-type module for a quantum group associated to an affine Kac-Moody algebra is characterized by its subspace of finite-dimensional weight spaces. In order to do this we prove an explicit equivalence of categories between a certain category containing the quantum Verma modules and a category of modules for a subalgebra of the quantum group for which the finite part of the Verma module is itself a module.

scholarly journals A construction of some ideals in affine vertex algebras

2003 ◽  
Vol 2003 (15) ◽  
pp. 971-980 ◽  
Author(s):  
Dražen AdamoviĆ
Keyword(s):  
Representation Theory ◽  
Lie Algebras ◽  
Vertex Operator ◽  
Operator Algebras ◽  
Vertex Operator Algebras ◽  
Affine Lie Algebras ◽  
Explicit Formulas ◽  
Singular Vectors ◽  
Weyl Algebras ◽  
New Family

We study ideals generated by singular vectors in vertex operator algebras associated with representations of affine Lie algebras of typesAandC. We find new explicit formulas for singular vectors in these vertex operator algebras at integer and half-integer levels. These formulas generalize the expressions for singular vectors from Adamović (1994). As a consequence, we obtain a new family of vertex operator algebras for which we identify the associated Zhu's algebras. A connection with the representation theory of Weyl algebras is also discussed.

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