The quantum general linear group again

Let V be an n-dimensional vector space over a field $\arbK$. Given an invertible $q\in\arbK$, let (y ,z) be the tortile Yang-Baxter-operator on V defined in an earlier section. By this example and this also , from an earlier section, there are strict tensor functors

$$\map M\from \tilBraid \to {\Mod_\arbK}\quad, \quad \map G\from \tilTan \to {\Mod_\arbK}$$

taking $\bigl(+\,, c_\plusplus\,, \thetaplus\bigr)$ to be $\bigl(V\mkern1mu ,y\,,z\bigr)$ (where we are identifying $\tilBraid$ with the subcategory of $\,\tilTan$ whose objects are positively signed sets, with arrows being ribbons which do not bend around).

Applying Tannaka Duality ideas to M and G, we obtain a co-balanced bi-algebra $E_{\!M}^\pprime$ and a co-tortile bi-algebra $\EsubG'$.

Theorem 19.1   There are $\arbK$-bialgebra isomorphisms (see example from an earlier section):

$$E_{\!M}^\pprime\;\isom\; \Mat_q(n)\quad, \quad \EsubG'\;\isom\; \GL_q(n)\;.$$

Corollary 19.2  

$M_{\!q}^{\phj}\!(n)$ is a co-balanced bi-algebra and $\GL_q(n)$ is a co-tortile bi-algebra.

The co-braiding given by $\map \gamma \from \GL_q(n) \XO \GL_q(n) \to \arbK$, and co-twist given by $\map \tau \from \GL_q(n) \XO \GL_q(n) \to \arbK$, satisfy the equations

$$\begin{eqnarray*} y(\epseps[i,j]) &\; = \; & \sum\limits_{m,r} \gamma(x_{im}^{... ...;\;\; &\; = \; & \sum\limits_m\tau(x_{im}^{\phj})\,\veps_m\hfill \end{eqnarray*}$$

This means:

$$\begin{eqnarray*} &\gamma(x_{im}^{\phj},x_{jr}^{\phj}) &\; = \; \cases{ \hfil 1... ...tau(x_{\!ij}^{\phj}) &\; = \qquad q^n_{}\,\delta_{ij}^{\phj}\;. \end{eqnarray*}$$