Gauge theories based on quantum groups, representations of the braid group
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Date
1993.
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Thesis (M.S.) - Bogazici University. Institute for Graduate Studies in Science and Engineering, 1993.
Abstract
Possible formulations of gauge field models where the gauge group is a quantum group are discussed. The exponential map from the generators of the Lie algebra analog of the quantum group SUq(2) to the quantum group SUq(2) itself is presented. The q-deformed Yang-Mills theory is introduced via the definition of the q-trace and the q-deformed YangMills lagrangian which is invariant under the quantum group gauge transformations. The gauge field takes values in the quantum universal enveloping algebra of SUq(2). As a result of this construction a Weinberg type mixing angle which depends on the quantum group deformation parameter q is obtained. The representations of the n-braid group where generators are given essentially by 2 x 2 matrices whose elements belong to a noncommutative algebra are presented. The Burau representation arises as a special (commuting) case of this algebra. A closely related algebra to the braid algebra is introduced and it is shown that the generalized oscillator system given by this algebra generates a hydrogen-like spectrum.