Given a real finite-dimensional vector space $V$ with a symmetric bilinear form $b$, we define the Clifford algebra $Cl(V,b)$ as the quotient of the tensor algebra $\bigotimes V$ by the two sided ideal generated by $v\otimes v + b(v,v) \mathbb{1}$ for all $v \in V$. $Cl(V,b)$ is a $\mathbb{Z}_2$-graded unital real associative algebra.

Every such $(V,b)$ is isomorphic to $\mathbb{R}^{r+s+t}$ with bilinear form defined by polarisation from the quadratic form $$ q(x) = \sum_{i=1}^{r + s + t} \varepsilon_i x_i^2, $$ where $$ \varepsilon_i = \begin{cases} 0 & i = 1,\dots,r\\ 1 & i = r+1,\dots,r+s\\ -1 & i = r+s+1, \dots, r+s+t .\end{cases} $$ Let $Cl(r,s,t)$ denote the Clifford algebra of $\mathbb{R}^{r+s+t}$ and the above bilinear form.

As $\mathbb{Z}_2$-graded unital real associative algebras, $$ Cl(r,s,t) \cong \Lambda \mathbb{R}^r \hat\otimes Cl(s,t), $$ where $\hat\otimes$ is the $\mathbb{Z}_2$-graded tensor product and where $Cl(s,t):= Cl(0,s,t)$ are the standard Clifford algebras associated to non-degenerate bilinear forms.

The representations of $Cl(s,t)$ are well-known: there are either one or two simple modules (up to isomorphism) depending on $s,t$ and every finite-dimensional module is a direct sum of simples.

I am interested in the representations of $Cl(r,s,t)$ for $r=1$, but more generally for $r>0$.

For $(s,t) = (1,0)$ and $(0,1)$, it is easy to work this out "by hand". The resulting category of representations is no longer semisimple, but it is not hard to show that any finite-dimensional module is a direct sum of indecomposable (but not simple) modules.

But before attempting to study the case of general $(s,t)$, I wonder whether there is some technology out there which can be brought to bear on this problem.

More concretely, I have a couple of


  1. Would a knowledge of the indecomposable modules of $\Lambda \mathbb{R}$ and $Cl(s,t)$ be sufficient to determine the indecomposable modules of their $\mathbb{Z}_2$-graded tensor product? If so, how?

  2. Is there a classification of indecomposable finite-dimensional modules of the exterior algebra $\Lambda \mathbb{R}^r$ for $r>1$? If so, where?

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    $\begingroup$ For Q1: Morita equivalence. For Q2: $\Lambda$-modules (and their derived category) were studied as part of Beilinson equivalence with coherent sheaves over $\Bbb{P}^n$. $\endgroup$ Nov 23 '18 at 16:44

To question 2: The exterior algebra is a finite dimensional local quiver algebra with loops $x_1,...,x_n$ and relations $x_i^2,x_i x_j + x_j x_i (i<j)$.

Local quiver algebras with at least 3 loops are always wild and thus a classification of their indecomposable modules is considered to be impossible, see for example http://www.math.uni-bonn.de/people/schroer/fd-atlas-files/FD-WildAlgebras.pdf for definitions of wild.

For $n=2$ the exterior algebra is tame and there should be a classification of all modules but I do not know an explicit reference at the moment. There is a trick to factor out the socle of the algebra and then being able to classifiy all representations via the classification of all representations of the Kroenecker algebra. This technique can be found in the book "Representations and Cohomoloy Volume 1" by Benson in chapter 4.3. where the classification is for the group algebra of the Klein Four group in characteristic 2, which has the same quiver but with relations $x_i^2,x_1 x_2 - x_2 x_1 $, so that the two algebras coincide in characteristic 2.


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