On The Representations Theory of Quantum Groups

Abstract

This work is concerned mainly with the structure theory of quantum groups and its representations, more precisely its induction of representa­ tions. Quantum groups were introduced since 1986 by V. G. Drinfeld [9] and investigated by S.Majid [15], and is still the subject of developments both in mathematics and theoretical physics. Such a structure, which is related to the structure of Hopf bialgebras and Yang Baxter equations take their origin in various fields of quasitriangular Hopf algebra, non commutative geome­ try, algebraic topology, Knot theory, Braid groups, integrable systems and conformal field theory. Hopf algebras are an exciting new generalization of ordinary groups, they have a rich mathematical structure and numerous roles in situations where or­ dinary groups are not adequate. In fact, the most familiar point of view about groups is as collections of transformations, which assumed invertible, and ev­ ery closed collection of invertible transformations is, inevitably a group, this is the role of groups as symmetries. Hopf algebra can act on objects. How­ ever, now the transformations are not all invertible. Instead, Hopf algebra have a weaker structure, called the antipode S, which provides a nonlocal linearized inverse. In other words, not individual elements but certain linear combinations are invertible. Remarkably, this weaker invertibility is all that is actually used in applications. An another point of view a Hopf algebra is an algebra for which the dual linear space of the algebra is also an algebra. The algebra structure on the d uallinear space is expressed in terms of the original algebra A as a coproduct /':, : A ---+ A 0 A Supplementing an algebra by a coproduct (forming a coalgebra) restores a kind of input-output symmetry to the system. (Hopf algebra were studied and investigated by Moss.E.Sweedler [22], and Eiichi ABE [1]). The question now is what is a quantum group ? To get a clear answer for this question, we consider the following very simple idea, when one has an axiom or condition for an algebraic structure one can relax it to be holds only up to some cocycle element which is then required to obey some important consistency conditions which we have to specify. The most important appli­ cation of this principle will be to the condition of cocommutitivity. Thus, we now study a class of Hopf algebras'•that are cocommutative only up to conjugation by an element R.