Unitary Representations for Twisted Product of Matrix Quantum Groups’

This paper is acontinuation of [JW], where we constructed afamily of compact matrix quantum groups in the sense ofWoronowicz [SLW2]. The construction followed the scheme provided by Woronowicz in [SLW3], in which the basic role is played by aproperly chosen function on permutations. In our case the function is related to counting the number of cycles in permutations. In [JW] we described the$.\mathrm{C}^{*}$-algebraic structure of the constructed objects. Here we shall concentrate on the “quantum group ” structure (Hopf algebra structure) and unitary representations of the quantum groups. As defined by Woronowicz in [SLW2], acompact matrix quantum group $(A, u) $ consists of aC’-algebra $A $ and an $N $ by $N $ matrix $u=(u_{jk})_{j,k=1}^{N} $ , with the elements $Ujk\in A$ generating a $\mathrm{d}\mathrm{e}\mathrm{n}\mathrm{s}\mathrm{e}*$-subalgebraA of $A $ , and with the following additional structure: 1. a $C^{*}$-homomorphism $\Phi $ : $Aarrow A\otimes A $ , called the $\mathrm{c}\mathrm{o}$-multiplication, such that $\Phi(u_{jk})=\sum_{r=0}^{N}u_{jr}\otimes u_{rk} $ (1.1) 2. alinear anti-multiplicative mapping $\kappa $ : $A $ $arrow A$, called the $\mathrm{c}\mathrm{o}$-inverse, such that $\kappa(\kappa(a^{*})^{*})=a $ for all elements $a\in A $ , and $\sum_{r=1}^{N}\kappa(u_{jr})u_{rk}=\delta_{jk}I $ (1.2