The $ SU(2)$ Quantum Group

Let us consider the structure of the ubiquitous quantum $ SU(2)$ group (Woronowicz 1987, Chaician and Demichev 1996). Here $ A$ is taken to be a C*-algebra generated by elements $ \alpha $ and $ \beta$ subject to the relations:

\begin{equation*}\begin{aligned}\alpha \alpha ^* + \mu^2 \beta \beta^* &= 1~,~ \...
...~ \alpha ^* \beta^* &= \mu^{-1} \beta^* \alpha ^*~, \end{aligned}\end{equation*}

where $ \mu \in [-1, 1]\backslash \{0\}$ . In terms of the matrix

$\displaystyle u = \bmatrix\alpha & - \mu \beta^* \\ \beta & \alpha ^* \endbmatrix$ (0.2)

the coproduct $ \Delta$ is then given via

$\displaystyle \Delta (u_{ij}) = \sum_k u_{ik} \otimes u_{kj}~.$ (0.3)

Example 0.1   The $ SL_q (2)$ Hopf algebra. The Hopf algebra $ SL_q (2)$ is defined by the generators $ a, b, c, d$ and the following relations:

$\displaystyle ba = qab~,~ db=qbd ~,~ ca = qac~,~dc = qcd~,bc = cb~,$ (0.4)

together with

$\displaystyle ad – da = (q^{-1}- q)bc ~,~ad – q^{-1}bc = 1~,$ (0.5)

and

$\displaystyle \Delta \bmatrix a & b \\ c & d \endbmatrix = \bmatrix a & b \\ c ...
...ix a & b \\ c & d \endbmatrix = \bmatrix d & -qb \\ -q^{-1}c & a \endbmatrix ~.$ (0.6)



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