Inductive constructions for Lie bialgebras and Hopf algebras

Abstract

In recent years, two generalisations of the theory of Lie algebras have become prominent, namely the "semi-classical" theory of Lie bialgebras and the "quantum" theory of Bopf algebras, including the quantized enveloping algebras. I develop an inductive approach to the study of these objects. An important tool is a construction called double-bosonisation defined by Majid for both Lie bialgebras and Hopf algebras, inspired by the triangular decomposition of a Lie algebra into positive and negative roots and a Cartan subalgebra. We describe two specific applications. The first uses double-bosonisation to add positive and negative roots and considers the relationship between two algebras when there is an inclusion of the associated Dynkin diagrams. In this setting, which we call Lie induction, doublebosonisation realises the addition of nodes to Dynkin diagrams. We use our methods to obtain necessary conditions for such an induction to be simple, using representation theory, providing a different perspective on the classification of simple Lie algebras. We consider the corresponding scheme for quantized enveloping algebras, based on inclusions of the associated root data. We call this quantum Lie induction. We prove that we have a double-bosonisation associated to these inclusions and investigate the structure of the resulting objects, which are Hopf algebras in braided categories, that is, covariant Bopf algebras. The second application generalises one of the most important constructions in this field, namely the Drinfel'd double of a Lie bialgebra, which has dimension twice that of the underlying algebra. Our construction, the triple, has dimension three times that of the input algebra. Our main result is that when the input algebra is factorisable, this is isomorphic to the triple direct sum as an algebra and a twisting as a coalgebra. We also indicate a number of ways in which the triple is related to the double.