Fix an alternating bicharacter $c$ as in (4.2). Then $c$ is a 2-cocycle on $\Gamma$ and

$$c(\alpha,\beta)c(\beta,\alpha)^{-1}=c(\alpha,\beta)^{2}=\sigma(\alpha,\beta)$$

for $\alpha,\beta\in\Gamma$ . As in (3.3), we write $k^{c}\Gamma$ in terms of a basis $\{x_{\alpha}\mid\alpha\in\Gamma\}$ such that $x_{\alpha}x_{\beta}=c(\alpha,\beta)x_{\alpha+\beta}$ for $\alpha,\beta\in\Gamma$ . Thus $A=k^{c}\Gamma^{+}$ equals the subspace of $k^{c}\Gamma$ spanned by the $x_{\alpha}$ for $\alpha\in\Gamma^{+}$ .

Similarly, we write the group algebra $k\Gamma$ in terms of a basis $\{y_{\alpha}\mid\alpha\in\Gamma\}$ , with $y_{\alpha}y_{\beta}=y_{\alpha+\beta}$ for $\alpha,\beta\in\Gamma$ , and we identify the semigroup algebra $R:=k\Gamma^{+}$ with the subspace of $k\Gamma$ spanned by the $y_{\alpha}$ for $\alpha\in\Gamma^{+}$ . We also view $R$ as a polynomial ring $k[y_{1},\dots,y_{n}]$ , as in Section 2, where $y_{i}=y_{\epsilon_{i}}$ and $\epsilon_{i}$ is the $i$ -th standard basis element of $\Gamma$ .

First, let $A$ be the standard one-parameter quantization ${\Cal{O}}_{q}(k^{n})$ , for some $q\in k^{\times}$ . Then $A$ is generated by $x_{1},\dots,x_{n}$ such that $x_{i}x_{j}=qx_{j}x_{i}$ for all $i<j$ . The alternating bicharacter $\sigma$ as in (4.1) can be expressed by

$$\sigma(\alpha,\beta)=q^{b(\alpha,\beta)},$$

where $b:\Gamma\times\Gamma\rightarrow{\mathbb{Z}}$ is the alternating bilinear form

$$b(\alpha,\beta)=\sum_{i<j}\alpha_{i}\beta_{j}-\sum_{i>j}\alpha_{i}\beta_{j}.$$

To complete the hypotheses of (4.11), we must assume that $q$ is either not a root of unity or an odd root of unity. In either case, $q$ has a square root $p\in k^{\times}$ such that $-1\notin\langle p\rangle$ (if $\operatorname{char}k\neq 2$ ), and the rule

$$c(\alpha,\beta)=p^{b(\alpha,\beta)}$$

defines an alternating bicharacter $c$ on $\Gamma$ satisfying the conclusions of (4.2). Hence, we can identify $A$ with $k^{c}\Gamma^{+}$ for this $c$ .