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Hopf-von Neumann algebra bicrossproducts, Kac algebra bicrossproducts, and the classical Yang-Baxter equations. (English) Zbl 0741.46033

From the text: “The idea of semidirect products of operator algebras by group actions has proven a very fruitful one for generating new examples.”
“Two groups \(G_ 1\), \(G_ 2\) are said to be a matched pair if each acts on the space of the other and these actions, \((\alpha,\beta)\) say, obey a certain compatibility condition. For every matched pair of locally compact groups \((G_ 1,G_ 2,\alpha,\beta)\) we construct an associated coinvolutive Hopf-von Neumann algebra \({\mathcal M}(G_ 1)^ \beta\bowtie_ \alpha L^ \infty(G_ 2)\) by simultaneous cross product and cross coproduct. For non-trivial \(\alpha\), \(\beta\) these bicrossproduct Hopf-von Neumann algebras are non-commutative and non- cocommutative. If the modules for the actions \(\alpha\), \(\beta\) 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 \(G_ 1\), \(G_ 2\) and \(\alpha\), \(\beta\) interchanged. Examples exist with \(G_ 1\) a simply connected Lie group and choices of \(G_ 2\) determined by suitable solutions of the classical Yang-Baxter equations on the complexification of the Lie algebra of \(G_ 1\).”.

MSC:

46L55 Noncommutative dynamical systems
46L87 Noncommutative differential geometry
47L50 Dual spaces of operator algebras
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