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

Shahn Majid

Vol. 141 (1990), No. 2, 311–332

DOI: 10.2140/pjm.1990.141.311

Abstract

Two groups G, H are said to be a matched pair if they act on each other and these actions, (α,β), obey a certain compatibility condition. In such a situation one may form a bicrossproduct group, denoted G_β⋈_αH. 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^∗), in a natural way.

As an example we construct the matched pair in detail in the case (SU(2),SU(2)^∗) where

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).

Mathematical Subject Classification 2000

Primary: 17B05

Secondary: 17B37, 57S25, 58F05

Milestones

Received: 4 March 1988

Revised: 6 December 1988

Published: 1 February 1990