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Title: Generalized quantum Gaudin spin chains, involutive automorphisms and 'twisted' classical r-matrices

Abstract

Using automorphisms {sigma} of finite-dimensional Lie algebras and classical skew-symmetric r-matrices we construct 'twisted' nonskew symmetric r-matrices with the spectral parameters. Using them and results of our previous papers we construct new quantum spin chains that generalize famous Gaudin spin chains. We consider several examples of the twisted nonskew symmetric r-matrices and the corresponding Gaudin-type systems.

Authors:
 [1]
  1. Bogoliubov Institute for Theoretical Physics, Institute of Mathematics of NASU, Metrologichna st.14-b, Kiev 03143 (Ukraine)
Publication Date:
OSTI Identifier:
20768747
Resource Type:
Journal Article
Resource Relation:
Journal Name: Journal of Mathematical Physics; Journal Volume: 47; Journal Issue: 3; Other Information: DOI: 10.1063/1.2179052; (c) 2006 American Institute of Physics; 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; QUANTUM MECHANICS; R MATRIX; SPIN

Citation Formats

Skrypnyk, T. Generalized quantum Gaudin spin chains, involutive automorphisms and 'twisted' classical r-matrices. United States: N. p., 2006. Web. doi:10.1063/1.2179052.
Skrypnyk, T. Generalized quantum Gaudin spin chains, involutive automorphisms and 'twisted' classical r-matrices. United States. doi:10.1063/1.2179052.
Skrypnyk, T. Wed . "Generalized quantum Gaudin spin chains, involutive automorphisms and 'twisted' classical r-matrices". United States. doi:10.1063/1.2179052.
@article{osti_20768747,
title = {Generalized quantum Gaudin spin chains, involutive automorphisms and 'twisted' classical r-matrices},
author = {Skrypnyk, T.},
abstractNote = {Using automorphisms {sigma} of finite-dimensional Lie algebras and classical skew-symmetric r-matrices we construct 'twisted' nonskew symmetric r-matrices with the spectral parameters. Using them and results of our previous papers we construct new quantum spin chains that generalize famous Gaudin spin chains. We consider several examples of the twisted nonskew symmetric r-matrices and the corresponding Gaudin-type systems.},
doi = {10.1063/1.2179052},
journal = {Journal of Mathematical Physics},
number = 3,
volume = 47,
place = {United States},
year = {Wed Mar 15 00:00:00 EST 2006},
month = {Wed Mar 15 00:00:00 EST 2006}
}
  • Using a one-parametric family of automorphisms {sigma}(u) of finite-dimensional simple Lie algebras g and classical skew-symmetric r-matrices r{sub 12}(u-v), we construct ''twisted'' nonskew symmetric classical r-matrices r{sub 12}{sup {sigma}}(u,v) with spectral parameters. We show that in the special case {sigma}(u)=Ad{sub K(u)} and classical matrix Lie algebras g, they are connected with the quasiclassical limit of the reflection equation algebras. Using them and results of our previous papers, we construct new integrable quantum spin chains of the ''generalized Gaudin'' type with an arbitrary value of spin living at each site.
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