dorsal/arxiv
View SchemaBoundary Solutions of the Classical Yang-Baxter Equation
| Authors | Murray Gerstenhaber, Anthony Giaquinto |
|---|---|
| Categories | |
| ArXiv ID | q-alg/9609014 |
| URL | https://arxiv.org/abs/q-alg/9609014 |
Abstract
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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"abstract": "We define a new class of unitary solutions to the classical Yang-Baxter\nequation (CYBE). These ``boundary solutions\u0027\u0027 are those which lie in the\nclosure of the space of unitary solutions to the modified classical Yang-Baxter\nequation (MCYBE). Using the Belavin-Drinfel\u0027d classification of the solutions\nto the MCYBE, we are able to exhibit new families of solutions to the CYBE. In\nparticular, using the Cremmer-Gervais solution to the MCYBE, we explicitly\nconstruct for all n \u003e 2 a boundary solution based on the maximal parabolic\nsubalgebra of sl(n) obtained by deleting the first negative root. We give some\nevidence for a generalization of this result pertaining to other maximal\nparabolic subalgebras whose omitted root is relatively prime to $n$. We also\ngive examples of non-boundary solutions for the classical simple Lie algebras.",
"arxiv_id": "q-alg/9609014",
"authors": [
"Murray Gerstenhaber",
"Anthony Giaquinto"
],
"categories": [
"q-alg",
"math.QA"
],
"title": "Boundary Solutions of the Classical Yang-Baxter Equation",
"url": "https://arxiv.org/abs/q-alg/9609014"
},
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