Quasi-{$*$} Structure on {$q$}-Poincare algebras

Details Quasi-{$*$} Structure on {$q$}-Poincare algebras Quasi-{$*$} Structure on {$q$}-Poincare algebras

1995-03-23

arxiv.org/abs/q-alg/9503014v1

Mathematics

Quantum Algebra

LATEX 56 pages and two figures

Scientific paper

We use braided groups to introduce a theory of $*$-structures on general inhomogeneous quantum groups, which we formulate as {\em quasi-$*$} Hopf algebras. This allows the construction of the tensor product of unitary representations up to a quantum cocycle isomorphism, which is a novel feature of the inhomogeneous case. Examples include $q$-Poincar\'e quantum group enveloping algebras in $R$-matrix form appropriate to the previous $q$-Euclidean and $q$-Minkowski spacetime algebras $R_{21}x_1x_2=x_2x_1R$ and $R_{21}u_1Ru_2=u_2R_{21}u_1R$. We obtain unitarity of the fundamental differential representations. We show further that the Euclidean and Minkowski Poincar\'e quantum groups are twisting equivalent by a another quantum cocycle.

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