Quasi-{$\ast$} Structure on {$q$}-Poincare algebras

Abstract

We use braided groups to introduce a theory of $\ast$-structures on general inhomogeneous quantum groups, which we formulate as {\em quasi-$\ast$} 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 $q$-Poincaré quantum group enveloping algebras in $R$-matrix form appropriate to the previous $q$-Euclidean and $q$-Minkowski spacetime algebras $R_{21}{x}_1{x}_2=x_2{x}_{1}R$ and $R_{21}{u}_{1}R{u}_2=u_2{R}_{21}{u}_{1}R$. We obtain unitarity of the fundamental differential representations. We show further that the Euclidean and Minkowski Poincaré quantum groups are twisting equivalent by a another quantum cocycle.

LATEX 56 pages and two figures

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