Representation Theory Of Hopf Algebras

Abstract

In this work we study and calculate some invariants of Hopf algebras . -~'" ...

The theory has been developed by Shahn Majidl l] and other workers such as N. Yu Reshetikhin and V.G Turaev. [5]. The actual calculation of the invariants from knot diagrams is still an expanding area of research.

In chapter 2 we develop the basic theory of Hopf algebras.Hopf algebras first appeared in the work of E. H. Hopf in connection with the cohomology of

4

groups. They also appeared in the work of G.I. Kac and co-workers in the study of group duals. e.g[6]. They have been used extensively by algebraic geometers. who traditionally work with the ring or algebra of functions on a space as a definition of the algebraic space. Hopf algebras have also traditionally been used as a tool in studying Lie algebras over a field k. A standard text is [2].

In chapter 3 we study objects loosely called quantum groups which are gen- eralizations of Hopf algebras. The notion of quantum groups is due to V.G Drinfeld [22] as an abstraction of structures implicit in the works of E.K. Sklyanin [7, 8] P. P. Kulish and N. Yu. Reshetikhin , M. Jimbo and others workingin quantum inverse scattering. They encountered certain symmetries that were later discovered to behave like Hopf algebras. The name 'quantum'

has no mathematical significance. It was a~-n..,._a.-m-

this new symmetry in quantum physics. ..

e used by physicists to denote

In chapter 4 we introduce the notion of 'deformation of a Hopf algebra' or

'twisting' constructions. 'Phis is the technique used to obtain new quantum groups from old ones. Deformation simply means making the Hopf algebra relations depend on some parameter(s). The basic theory was developed by

Drinfeld [9, 10, 11]. The theory of co cycles and cohomology was developed by M.E sweedler [13]

In chapter 5 we develop a conceptual framework for the study of the rep- resentations of Hopf algebras. We use the language of category theory. A standard text is [3] of S. Mac lane. Braided monoidal categories were formally introduced into category theory by A. Joyal and R. Street [12]. They also arose in the study of the representation theory of quantum groups shortly after Drinfeld's seminal work [4]. A systematic treatment was developed by Shahn Majid [1] where they are called quasi tensor categories. The notion of categorical dimension and trace in braided categories was developed by Shahn Majid as a generalization of the rank in symmetric monoidal categories [1] where he also calculated the invariant on the figure of eight knot. New in- variants are calculated for knots with up Yl_Iline double crossing points . ..

1.2 Objectives

~.

This research aims to summarise the major concepts relevant to the calcula- bon of Hopf algebra invariants. We show how to obtain new algebraic struc-

tures from old ones using non-abelian cohomology. Finally we generalise the notion of categorical dimension and calculate Hopf algebra invariants using prime knots of up to nine crossings.

1.3 Methodology

The reference material came from books, journals, periodicals and material from the World Wide Web. We use the traditional mathematical definition- theorem-proof sequence.

1.4 Significance

This work is part of the continuing efforts-by- algebraists and knot theorists aimed at a fuller understanding of the relationship between knot theory and quantum group representations.