Inductive constructions for Lie bialgebras and Hopf algebras
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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.
Authors
Grabowski, Jan E.Collections
- Theses [3834]