Nonstandard quantum groups: twisting constructions and noncommutative differential geometry

Abstract

The general subject of this thesis is quantum groups. The major original results are obtained in the particular areas of twisting constructions and noncommutative differential geometry.

Chapters 1 and 2 are intended to explain to the reader what are quantum groups. They are written in the form of a series of linked results and definitions. Chapter 1 reviews the theory of Lie algebras and Lie groups, focusing attention in particular on the classical Lie algebras and groups. Though none of the quoted results are due to the author, such a review, aimed specifically at setting up the paradigm which provides essential guidance in the theory of quantum groups, does not seem to have appeared already. In Chapter 2 the elements of the quantum group theory are recalled. Once again, almost none of the results are due to the author, though in Section 2.10, some results concerning the nonstandard Jordanian group are presented, by way of a worked example, which have not been published.

Chapter 3 concerns twisting constructions. We introduce a new class of 2-cocycles defined explicitly on the generators of certain multiparameter standard quantum groups. These allow us, through the process of twisting the familiar standard quantum groups, to generate new as well as previously known examples of non-standard quantum groups. In particular we are able to construct generalisations of both the Cremmer-Gervais deformation of SL(3) and the so called esoteric quantum groups of Fronsdal and Galindo in an explicit and straightforward manner.

In Chapter 4 we consider the differential calculus on Hopf algebras as introduced by Woronowicz. We classify all 4-dimensional first order bicovariant calculi on the Jordanian quantum group GL[sub]h,[sub]g(2) and all 3-dimensional first order bicovariant calculi on the Jordanian quantum group SL[sub]h(2). In both cases we assume that the bicovariant bimodules are generated as left modules by the differentials of the quantum group generators. It is found that there are 3 1-parameter families of 4-dimensional bicovariant first order calculi on GL[sub]h,[sub]g(2) and that there is a single, unique, 3-dimensional bicovariant calculus on SL[sub]h(2). This 3-dimensional calculus may be obtained through a classical-like reduction from any one of the three families of 4-dimensional calculi on GL[sub]h,[sub]g(2). Details of the higher order calculi and also the quantum Lie algebras are presented for all calculi. The quantum Lie algebra obtained from the bicovariant calculus on SL[sub]h(2) is shown to be isomorphic to the quantum Lie algebra we obtain as an ad-submodule within the Jordanian universal enveloping algebra U[sub]h(sl[sub]2(C)) and also through a consideration of the decomposition of the tensor product of two copies of the deformed adjoint module. We also obtain the quantum Killing form for this quantum Lie algebra.