Double cross biproduct and bi-cycle bicrossproduct Lie bialgebras
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. \textit{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 \textit{A. Masuoka}'s construction of cross product Lie bialgebras [Trans. Am. Math. Soc. 352, No. 8, 3837--3879 (2000; Zbl 0960.16045)].
scientific article; zbMATH DE number 5780957
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