Noncommutative phase and the unitarization of the quantum group GL(2)

Abstract

In this thesis, a new *-operation (or unitarized form) is defined for the two-parameter quantum group GLp,q(2) in the case that pq is real, and the new group is denoted by Up,q(2). The most interesting aspect of our construction is the appearance of the noncommutative phase described by the unitary operator u. The operator u with a central hermitian operator s allow us to extend the algebra of the quantum group GLp,q(2) in order to obtain not only the *-operation but also the *-relations throughout the new algebra of Up,q(2). It is shown how certain *-representations of the quantum group SUq(2) can be extended in order to give *-representations of Up,q(2). This not only allows us to verify the algebraic relations in a representation, it also gives a hint of possible physical interpretations of the algebraic generators as operators.