Invariant theory in exterior algebras and Amitsur-Levitzki type theorems

Abstract

This article discusses invariant theories in some exterior algebras, which are closely related to Amitsur-Levitzki type theorems. First we consider the exterior algebra on the vector space of square matrices of size $n$, and look at the invariants under conjugations. We see that the algebra of these invariants is isomorphic to the exterior algebra on an $n$-dimensional vector space. Moreover we give a Cayley-Hamilton type theorem for these invariants (the anticommutative version of the Cayley-Hamilton theorem). This Cayley-Hamilton type theorem can also be regarded as a refinement of the Amitsur-Levitzki theorem. We discuss two more Amitsur-Levitzki type theorems related to invariant theories in exterior algebras. One is a famous Amitsur-Levitzki type theorem due to Kostant and Rowen, and this is related to $O(V)$-invariants in $\Lambda(\Lambda_2(V))$. The other is a new Amitsur-Levitzki type theorem, and this is related to $GL(V)$-invariants in $\Lambda(\Lambda_2(V) \oplus S_2(V^*))$.