Matched pairs of Lie groups associated to solutions of the Yang-Baxter equations

Abstract

Two groups G, H are said to be a matched pair if they act on each other and these actions, (a, /?), obey a certain compatibility condition.In such a situation one may form a bicrossproduct group, denoted Gβ cχi Q H. Also in this situation one may form a bicrossproduct Hopf, Hopf-von Neumann or Kac algebra obtained by simultaneous cross product and cross coproduct.We show that every compact semi-simple simply-connected Lie group G is a member of a matched pair, denoted (G, G*) 9 in a natural way.As an example we construct the matched pair in detail in the case (SU(2), SU( 2)*) where SU(2)* = is the simply-connected group of a Lie algebra su(2)*.Here su(2)* is defined with respect to a standard canonical solution of the CYBE on the complexification of su(2).