## Double cross biproduct and bi-cycle bicrossproduct Lie bialgebras.(English)Zbl 1203.17011

The author considers various constructions of Lie bialgebras. First he reviews Majid’s construction starting from Lie bialgebras $$A$$ and $$H$$ with $$(A,H)$$ a matched pair of Lie algebras, $$A$$ a left H-module Lie coalgebra and $$H$$ a right A-module Lie coalgebra with an additional technical condition. S. Majid then constructed a double crossed product Lie bialgebra structure on $$A+H$$ [Foundations of quantum groups, Cambridge Univ. Press (1995; Zbl 0857.17009)]. In the paper under review, the author gives an analogous construction for matched pairs of Lie coalgebras, calling the result a double cross coproduct Lie bialgebra. The first main theorem considers a matched pair $$(A,H)$$ of Lie algebras and Lie coalgebras, $$A$$ a braided Lie bialgebra in the category of left Yetter-Drinfeld modules over $$H, H$$ a braided Lie bialgebra in the category of right Yetter-Drinfeld modules over $$A$$. Out of this comes a double cross biproduct of $$A$$ and $$H$$, which is a Lie bialgebra. This generalizes Majid’s construction cited above. The other main theorem constructs a bi-cycle bicrossproduct Lie bialgebra. Here $$A$$ and $$H$$ are Lie algebras and Lie coalgebras, where various module and comodule structures relate $$A$$ and $$H$$, there are cocycles from $$H$$ tensor $$H$$ to $$A$$ and from $$A$$ tensor $$A$$ to $$H$$ yielding a cycle cross product, cycles from $$A$$ to $$H$$ tensor $$H$$ and $$H$$ to A tensor A and a cocycle from $$A$$ tensor $$A$$ to $$H$$ yielding a cycle cross coproduct. Assuming some technical conditions, all this fits together to form a Lie bialgebra called a bi-cycle bicrossproduct Lie bialgebra. This generalizes A. Masuoka’s construction of cross product Lie bialgebras [Trans. Am. Math. Soc. 352, No. 8, 3837–3879 (2000; Zbl 0960.16045)].

### MSC:

 17B62 Lie bialgebras; Lie coalgebras

### Keywords:

Lie bialgebra; crossed product

### Citations:

Zbl 0960.16045; Zbl 0857.17009
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