Grassmann Algebras

The Grassmann algebra $G$ is the algebra over an area $\mathbb{F}$ created by the variables $e_i$ such that $e_i^2=0$ and also $e_i e_j = - e_j e_i$.

I'm seeking some referrals on algebras $G \otimes A$ (In this instance, the Grassmann algebra mores than $\mathbb{C}$) where $A$ is the team algebra $\mathbb {C} \mathbb{Z}_n$ created by $U_t$ and also $e_i U_t = \zeta U_t e_i$ where $\zeta$ is a $n$ - th primitive origin of unity. Or to put it simply, I intend to attach an additional variable $t$ to the Grassmann algebra such that $t^n=1$ and also $e_i t = \zeta t e_i $.

The factor I ask this, is that I found algebras that have the very same polynomial identifications as the ones over, so I intend to see what can I claim concerning them from just the identifications.

I assume since these algebras are matrix algebras with coefficients in $\mathbb{Q}(\zeta)G$ such that each diagonal is increased by a (solitary) component 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 $g_0, g_2 \in G_0$ (also) and also $g_1, g_3 \in G_1$ (weird). The $t$ variable is the angled matrix with coefficients $1,i,-1,-i$.

Many thanks.

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2019-12-02 02:51:57
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