Complex Quantum Groups and Their Real Representations

Abstract

We define $\ast$-Hopf algebras $Fun({SL}_q(N,C;{\epsilon }_1,\dots ,{\epsilon }_N))$, $Fun(O_q(N,C;{\epsilon }_1,\dots ,{\epsilon }_N))$ and $Fun({Sp}_q(n,C;{\epsilon }_1,\dots ,{\epsilon }_{2n}))$ as the real complexifications of $\ast$-Hopf algebras $Fun({SU}_q(N,C;{\epsilon }_1,\dots ,{\epsilon }_N))$, $Fun(O_q(N,C;{\epsilon }_1,\dots ,{\epsilon }_N))$ and $Fun({Sp}_q(N,C;{\epsilon }_1,\dots ,{\epsilon }_{2n}))$ of [RTF] (for $q>0$). Such construction can be done for each coquasitriangular (CQT) $\ast$-Hopf algebra $\mathbf{A}$. The obtained object ${\mathbf{A}}^{CR}$ is also a CQT $\ast$-Hopf algebra. We describe the theory of corepresentations of ${\mathbf{A}}^{CR}$ in terms of such a theory for $\mathbf{A}$.