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.

Levi and Malcev Theorems for Finite-Dimensional Algebras from the Variety Defined by the Identities x2 = J( x,y, zu) =0

2021 ◽  
Vol 28 (01) ◽  
pp. 87-90
Author(s):  
Óscar Guajardo Garza ◽  
Marina Rasskazova ◽  
Liudmila Sabinina
Keyword(s):  
Lie Algebras ◽  
Present Article ◽  
Variety Of Algebras ◽  
Finite Dimensional ◽  
Finite Dimensional Algebras

We study the variety of binary Lie algebras defined by the identities [Formula: see text], where [Formula: see text] denotes the Jacobian of [Formula: see text], [Formula: see text], [Formula: see text]. Building on previous work by Carrillo, Rasskazova, Sabinina and Grishkov, in the present article it is shown that the Levi and Malcev theorems hold for this variety of algebras.

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 ON THE SPECTRA OF ${{\widehat{sl}\left( N \right)_k } \mathord{\left/ {\vphantom {{\widehat{sl}\left( N \right)_k } {\widehat{sl}\left( N \right)_k }}} \right. \kern-\nulldelimiterspace} {\widehat{sl}\left( N \right)_k }}$ AND WN GRAVITIES

1993 ◽  
Vol 08 (29) ◽  
pp. 5115-5128
Author(s):  
VLADIMIR SADOV
Keyword(s):  
Tensor Product ◽  
Lie Algebras ◽  
Hamiltonian Reduction ◽  
Affine Lie Algebras ◽  
Brst Cohomology ◽  
Boundary States ◽  
One To One ◽  
Coset Models

We study the spectra of G/G coset models by computing BRST cohomology of affine Lie algebras with coefficients in tensor product of two modules. One-to-one correspondence between the spectra of [Formula: see text] and that of the minimal matter coupled to gravity (including boundary states of the Kac table) is observed. This phenomenon is discussed from the point of Hamiltonian reduction of BRST complexes of [Formula: see text] Lie algebras.

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 VERTEX ALGEBRAS ASSOCIATED WITH ELLIPTIC AFFINE LIE ALGEBRAS

2011 ◽  
Vol 13 (04) ◽  
pp. 579-605 ◽  
Author(s):  
JIANCAI SUN ◽  
HAISHENG LI
Keyword(s):  
Lie Algebras ◽  
Vertex Algebras ◽  
Affine Lie Algebras ◽  
Quantum Vertex ◽  
Special Case

We associate what we call vertex ℂ((z))-algebras and their modules in a certain category with elliptic affine Lie algebras. To a certain extent, this association is similar to that of vertex algebras and their modules with affine Lie algebras. While the notion of vertex ℂ((z))-algebra is a special case of that of quantum vertex ℂ((z))-algebra, which was introduced and studied by one of us (Li), here we use those results on quantum vertex ℂ(z))-algebras in an essential way. In the course of this work, we also construct and exploit two families of Lie algebras which are closely related to elliptic affine Lie algebras.

Integrable two-dimensional dynamical systems and the characteristic Lie algebras

2007 ◽  
Vol 5 ◽  
pp. 195-200
Author(s):  
A.V. Zhiber ◽  
O.S. Kostrigina
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Lie Algebra ◽  
Lie Algebras ◽  
Second Order ◽  
System Of Equations ◽  
Two Dimensional ◽  
Finite Dimensional ◽  
Dimensional Dynamical System

In the paper it is shown that the two-dimensional dynamical system of equations is Darboux integrable if and only if its characteristic Lie algebra is finite-dimensional. The class of systems having a full set of fist and second order integrals is described.

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