## 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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### References:

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