BOUNDARY SOLUTIONS OF THE CLASSICAL YANG-BAXTER EQUATION

Citation
M. Gerstenhaber et A. Giaquinto, BOUNDARY SOLUTIONS OF THE CLASSICAL YANG-BAXTER EQUATION, letters in mathematical physics, 40(4), 1997, pp. 337-353
Citations number
13
Categorie Soggetti
Mathematical Method, Physical Science","Physycs, Mathematical
ISSN journal
03779017
Volume
40
Issue
4
Year of publication
1997
Pages
337 - 353
Database
ISI
SICI code
0377-9017(1997)40:4<337:BSOTCY>2.0.ZU;2-M
Abstract
We define a new class of unitary solutions to the classical Yang-Baxte r equation (CYBE). These 'boundary solutions' are those which lie in t he closure of the space of unitary solutions of the modified classical Yang-Baxter equation (MCYBE). Using the Belavin-Drinfel'd classificat ion of the solutions to the MCYBE, we are able to exhibit new families of solutions to the CYBE. In particular, using the Cremmer-Gervais so lution to the MCYBE, we explicitly construct for all n greater than or equal to 3 a boundary solution based on the maximal parabolic subalge bra of sl(n) obtained by deleting the first negative root. We give som e evidence for a generalization of this result pertaining to other max imal parabolic subalgebras whose omitted root is relatively prime to n . We also give examples of nonboundary solutions for the classical sim ple Lie algebras.