\[lem2-1\] Let $R$ be an algebra, and $G=\{g_1,\dots,g_{n-1},g_n=1\}$ a finite group acting on $R$. Assume that $a_1,\dots,a_{n-1}\in R$ are central elements of $R$. Then the following two sequences $$\bigcup_{s=0}^{n-1}\left(g_{\widehat{[i_1i_2\cdots i_s]}}(a_{\widehat{[i_1i_2\cdots i_s]}}):1\leq i_1<\cdots< i_s\leq n-1\right)$$ and $$\bigcup_{s=0}^{n-1}((-1)^sa_{[i_1i_2\cdots i_s]}:1\leq i_1<\cdots< i_s\leq n-1)$$ are pertinent, where $a_{[\ ]}=1$ and $a_{\widehat{[\ ]}}=a_1a_2\cdots a_{n-1}$ when $s=0$.

For every $1\neq g\in G$, we have $$\sum_{[i_1i_2\cdots i_s]} g_{\widehat{[i_1i_2\cdots i_s]}}(a_{\widehat{[i_1i_2\cdots i_s]}})g(a_{[i_1i_2\cdots i_s]})= \prod_{i=1}^{n-1}(g_i(a_i)-g(a_i)),$$ where on the left hand side, the sum is over all the possible choices of the arrays $[i_1i_2\cdots i_s]$ for $s=0,\dots,n-1$. Since $g$ must be one of the elements in $\{g_1,\dots, g_{n-1}\}$, the right hand side of the above equation is zero. Hence the result follows from the definition of the pertinent sequences.

Now let $R$ be an algebra, and let $a_1,\dots,a_{n-1}\in R$ such that $a_ia_j=q_{ij}a_ja_i$ for all $i< j$ and some $q_{ij}\neq 0$. Let $G=\{g_1,\dots,g_{n-1},g_n=1\}$. Assume $g_i(a_i)=\xi_i a_i$ for $i=1,\dots,n-1$ with $\xi_i\in\kk$. Write $$q_{[i_1i_2\cdots i_s]}=\prod_{k=1}^s\prod_{\varepsilon} q_{i_kj},$$ where the restriction $\varepsilon$ under the second $\prod$ reads as $\{j>i_k,\hbox{ and }j\neq i_1,\dots, i_s\}$.

\[lem2-2\] With the assumptions as above, we have the following pertinent sequences $$\bigcup_{s=0}^{n-1}g_{\widehat{[i_1i_2\cdots i_s]}}(a_{\widehat{[i_1i_2\cdots i_s]}}:1\leq i_1<\cdots<i_s\leq n-1)$$ and $$\bigcup_{s=0}^{n-1}((-1)^sq_{[i_1i_2\cdots i_s]}a_{[i_1i_2\cdots i_s]}:1\leq i_1<\cdots<i_s\leq n-1),$$ where $a_{[\ ]}=1$ and $a_{\overline{[\ ]}}=a_1a_2\cdots a_{n-1}$ when $s=0$.

For 