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Good day all. I am wondering about a possible classification result.

Consider the Grassman algebra $\bigwedge V$ for a vector space $V$ of finite dimension. Is there a simple characterization of the simple modules over $\bigwedge V$? As in, how can we describe them all up to isomorphism? I did not see this result in my books, so I thought I ask here. Thank you.

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up vote 2 down vote accepted

As noted here: Are projective modules over exterior algebras of vector spaces necessarily free? the exterior algebra is local, and hence has only one maximal right ideal, call it M. Any simple module must therefore be isomorphic to the right $R$ module $\bigwedge(V)/M$.

Similarly, the simple left modules are isomorphic to the left $R$ module $\bigwedge(V)/M$.

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Thanks you for this. – Camilla Vaernes Apr 28 '12 at 2:03

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