HOPF-VONNEUMANN ALGEBRA BICROSS-PRODUCTS, KAC ALGEBRA BICROSS-PRODUCTS, AND THE CLASSICAL YANG-BAXTER EQUATIONS

Two groups G1, G2 are said to be a matched pair if each acts on the space of the other and these actions, (α, β) say, obey a certain compatibility condition. For every matched pair of locally compact groups (G1, G2, α, β) we construct an associated coinvolutive Hopf-von Neumann algebra M(G1)β{bowtie}αL∞(G2) by simultaneous cross product and cross coproduct. For non-trivial α, β these bicrossproduct Hopf-von Neumann algebras are non-commutative and non-cocommutative. If the modules for the actions α, β are also matched then these bicrossproducts are Kac algebras. In this case we show that the dual Kac algebra is of the same form with the roles of G1, G2 and of α, β interchanged. Examples exist with G1 a simply connected Lie group and choices of G2 determined by suitable solutions of the Classical Yang-Baxter Equations on the complexification of the Lie algebra of G1. © 1991.