Double cross biproduct and bi-cycle bicrossproduct Lie bialgebras
type: Item
instance of: scholarly article
title: Double cross biproduct and bi-cycle bicrossproduct Lie bialgebras
zbMATH Open document ID: 1203.17011
author: Tao Zhang
published in: Journal of Generalized Lie Theory and Applications
publication date: 2010-09-08T00:00:00Z
review text: 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)].
reviewed by: Earl J. Taft
Mathematics Subject Classification ID: 17B62
zbMATH DE Number: 5780957
zbMATH Keywords: Lie bialgebra
zbMATH Keywords: crossed product
MaRDI profile type: Publication
DOI: 10.4303/JGLTA/S090602
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description: scientific article; zbMATH DE number 5780957