Manin-Olshansky Triples for Lie Superalgebras

DOI

10.2991/jnmp.2000.7.2.4How to use a DOI?

Abstract

Following V. Drinfeld and G. Olshansky, we construct Manin triples (g, a, a ) such that g is different from Drinfeld's doubles of a for several series of Lie superalgebras a which have no even invariant bilinear form (periplectic, Poisson and contact) and for a remarkable exception. Straightforward superization of suitable Etingof­Kazhdan's results guarantee then the uniqueness of q-quantization of our Lie bialgebras. Our examples give solutions to the quantum Yang-Baxter equation in the cases when the classical YB equation has no solutions. To find explicit solutions is a separate (open) problem. It is also an open problem to list (à la Belavin-Drinfeld) all solutions of the classical YB equation for the Poisson superalgebras po(0|2n) and the exceptional Lie superalgebra k(1|6) which has a Killing-like supersymmetric bilinear form but no Cartan matrix.

Copyright

© 2006, the Authors. Published by Atlantis Press.

Open Access

This is an open access article distributed under the CC BY-NC license (http://creativecommons.org/licenses/by-nc/4.0/).

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TY  - JOUR
AU  - Dimitry Leites
AU  - Alexander Shapovalov
PY  - 2000
DA  - 2000/05/01
TI  - Manin-Olshansky Triples for Lie Superalgebras
JO  - Journal of Nonlinear Mathematical Physics
SP  - 120
EP  - 125
VL  - 7
IS  - 2
SN  - 1776-0852
UR  - https://doi.org/10.2991/jnmp.2000.7.2.4
DO  - 10.2991/jnmp.2000.7.2.4
ID  - Leites2000
ER  -