Manin-Olshansky triples for Lie superalgebras

Abstract

Following V. Drinfeld and G. Olshansky, we construct Manin triples $(\fg, \fa, \fa^*)$ such that $\fg$ is different from Drinfeld's doubles of $\fa$ for several series of Lie superalgebras $\fa$ which have no even invariant bilinear form (periplectic, Poisson and contact) and for a remarkable exception. Straightforward superization of suitable Etingof--Kazhdan'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 {\it classical} YB equation for the Poisson superalgebras $\fpo(0|2n)$ and the exceptional Lie superalgebra $\fk(1|6)$ which has a Killing-like supersymmetric bilinear form but no Cartan matrix.