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Title: Affine Kac-Moody symmetric spaces related with A{sub 1}{sup (1)}, A{sub 2}{sup (1)}, A{sub 2}{sup (2)}

Abstract

Symmetric spaces associated with Lie algebras and Lie groups which are Riemannian manifolds have recently got a lot of attention in various branches of Physics for their role in classical/quantum integrable systems, transport phenomena, etc. Their infinite dimensional counter parts have recently been discovered which are affine Kac-Moody symmetric spaces. In this paper we have (algebraically) explicitly computed the affine Kac-Moody symmetric spaces associated with affine Kac-Moody algebras A{sub 1}{sup (1)},A{sub 2}{sup (1)},A{sub 2}{sup (2)}. We hope these types of spaces will play similar roles as that of symmetric spaces in many physical systems.

Authors:
;  [1]
  1. Department of Mathematics, National Institute of Technology Rourkela, Rourkela 769008, Odisha (India)
Publication Date:
OSTI Identifier:
22306075
Resource Type:
Journal Article
Resource Relation:
Journal Name: Journal of Mathematical Physics; Journal Volume: 55; Journal Issue: 8; Other Information: (c) 2014 AIP Publishing LLC; Country of input: International Atomic Energy Agency (IAEA)
Country of Publication:
United States
Language:
English
Subject:
71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; ALGEBRA; LIE GROUPS; MATHEMATICAL MANIFOLDS; MATHEMATICAL SPACE; QUANTUM MECHANICS; RIEMANN SPACE; SYMMETRY

Citation Formats

Nayak, Saudamini, E-mail: anumama.nayak07@gmail.com, and Pati, K. C., E-mail: kcpati@nitrkl.ac.in. Affine Kac-Moody symmetric spaces related with A{sub 1}{sup (1)}, A{sub 2}{sup (1)}, A{sub 2}{sup (2)}. United States: N. p., 2014. Web. doi:10.1063/1.4889796.
Nayak, Saudamini, E-mail: anumama.nayak07@gmail.com, & Pati, K. C., E-mail: kcpati@nitrkl.ac.in. Affine Kac-Moody symmetric spaces related with A{sub 1}{sup (1)}, A{sub 2}{sup (1)}, A{sub 2}{sup (2)}. United States. doi:10.1063/1.4889796.
Nayak, Saudamini, E-mail: anumama.nayak07@gmail.com, and Pati, K. C., E-mail: kcpati@nitrkl.ac.in. Fri . "Affine Kac-Moody symmetric spaces related with A{sub 1}{sup (1)}, A{sub 2}{sup (1)}, A{sub 2}{sup (2)}". United States. doi:10.1063/1.4889796.
@article{osti_22306075,
title = {Affine Kac-Moody symmetric spaces related with A{sub 1}{sup (1)}, A{sub 2}{sup (1)}, A{sub 2}{sup (2)}},
author = {Nayak, Saudamini, E-mail: anumama.nayak07@gmail.com and Pati, K. C., E-mail: kcpati@nitrkl.ac.in},
abstractNote = {Symmetric spaces associated with Lie algebras and Lie groups which are Riemannian manifolds have recently got a lot of attention in various branches of Physics for their role in classical/quantum integrable systems, transport phenomena, etc. Their infinite dimensional counter parts have recently been discovered which are affine Kac-Moody symmetric spaces. In this paper we have (algebraically) explicitly computed the affine Kac-Moody symmetric spaces associated with affine Kac-Moody algebras A{sub 1}{sup (1)},A{sub 2}{sup (1)},A{sub 2}{sup (2)}. We hope these types of spaces will play similar roles as that of symmetric spaces in many physical systems.},
doi = {10.1063/1.4889796},
journal = {Journal of Mathematical Physics},
number = 8,
volume = 55,
place = {United States},
year = {Fri Aug 15 00:00:00 EDT 2014},
month = {Fri Aug 15 00:00:00 EDT 2014}
}
  • Simple procedures are given for finding all the dominant weights in a highest weight representation of an affine algebra, for finding the Weyl orbit of an arbitrary weight, and for determining whether or not any given weight is in any given representation. A simple definition of congruency is given that applies to all affine algebras. The standard indefinite scalar product is generalized; the generalization is used in the procedures.
  • Affine generalizations of some familiar notions from the representation theory of semisimple Lie algebras/groups are introduced, described and illustrated. The multiplicity of a weight and the dimension congruence class, and indices of a representation are touched upon. Examples of the highest weight representations of affine E/sub 8/ are considered as a preview of far more extensive results of this type to appear.
  • The determinant of twisted bosons related the Kac-Moody algebra is obtained. The correlation function of twist fields is also calculated under some assumptions.
  • We define a kind of quantized enveloping algebra of a generalized Kac-Moody algebra G by adding a generator J satisfying J{sup m}=J{sup m-1} for some integer m. We denote this algebra by wU{sub q}{sup {tau}}(G). This algebra is a weak Hopf algebra if and only if m=2. In general, it is a bialgebra, and contains a Hopf subalgebra. This Hopf subalgebra is isomorphic to the usually quantum envelope algebra U{sub q}(G) of a generalized Kac-Moody algebra G.
  • In this report, some important facts on the symmetries and conservation laws of high dimensional integrable systems are discussed. It is summarized that almost all the known (2+1)-dimensional integrable models possess the Kac-Moody-Virasoro (KMV) symmetry algebras. One knows that infinitely many partial differential equations may possess a same KMV symmetry algebra. It is found that the KMV symmetry groups can be explicitly obtained by using some direct methods. For some quite general variable coefficient nonlinear systems, their sufficient and necessary condition for the existence of the KMV symmetry algebra is they can be changed to the related known constant coefficientmore » models. Finally, it is found that every one symmetry may be related to infinitely many conservation laws and then infinitely many models may possess a same set of infinitely many conservation laws.« less