# Grassmann Algebras

The Grassmann algebra $G$ is the algebra over a field $\mathbb{F}$ generated by the variables $e_i$ such that $e_i^2=0$ and $e_i e_j = - e_j e_i$.

I'm looking for some references on algebras $G \otimes A$ (In this case, the Grassmann algebra is over $\mathbb{C}$) where $A$ is the group algebra $\mathbb {C} \mathbb{Z}_n$ generated by $U_t$ and $e_i U_t = \zeta U_t e_i$ where $\zeta$ is a $n$-th primitive root of unity. Or in other words, I want to attach another variable $t$ to the Grassmann algebra such that $t^n=1$ and $e_i t = \zeta t e_i$.

The reason I ask this, is that I came across algebras that have the same polynomial identities as the ones above, so I want to see what can I say about them from only the identities.

I think now that these algebras are matrix algebras with coefficients in $\mathbb{Q}(\zeta)G$ such that each diagonal is multiplied by a (single) element from $G$. Something like

$$\left(\begin{array}{rrrr} g_0 & g_1 & g_2 & g_3 \\ g_3 & g_0 & g_1 & g_2 \\ g_2 & g_3 & g_0 & g_1\\ g_1 & g_2 & g_3 & g_0 \end{array}\right)$$

where $g_0, g_2 \in G_0$ (even) and $g_1, g_3 \in G_1$ (odd). The $t$ variable is the diagonal matrix with coefficients $1,i,-1,-i$.

Thanks.

• You would increase your chances of getting a useful answer if you asked a crisper question: what do you want to know about these algebras? (Also, in one place you have an arbitrary field $\mathbb{F}$ and in another you have $\mathbb{C}$. Is this really what you mean? It seems more likely that you want them both to be the same.) – Pete L. Clark Nov 28 '10 at 0:15
• @Pete: edited the question – Ofir Nov 28 '10 at 6:44