Manin-Olshansky Triples for Lie Superalgebras
- 10.2991/jnmp.2000.7.2.4How to use a DOI?
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.
- © 2006, the Authors. Published by Atlantis Press.
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Cite this article
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 -