Boundary Solutions of the Classical Yang-Baxter Equation

Details Boundary Solutions of the Classical Yang-Baxter Equation Boundary Solutions of the Classical Yang-Baxter Equation

1996-09-16

arxiv.org/abs/q-alg/9609014v1

Mathematics

Quantum Algebra

21 pages, AMSTEX, A version of this report will appear in Lett. Math. Physics

Scientific paper

We define a new class of unitary solutions to the classical Yang-Baxter equation (CYBE). These ``boundary solutions'' are those which lie in the closure of the space of unitary solutions to the modified classical Yang-Baxter equation (MCYBE). Using the Belavin-Drinfel'd classification of the solutions to the MCYBE, we are able to exhibit new families of solutions to the CYBE. In particular, using the Cremmer-Gervais solution to the MCYBE, we explicitly construct for all n > 2 a boundary solution based on the maximal parabolic subalgebra of sl(n) obtained by deleting the first negative root. We give some evidence for a generalization of this result pertaining to other maximal parabolic subalgebras whose omitted root is relatively prime to $n$. We also give examples of non-boundary solutions for the classical simple Lie algebras.

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