Back to the Amitsur–Levitzki Theorem: a Super Version for the Orthosymplectic Lie Superalgebra \frak o s p(1, 2n)

Abstract

We prove an Amitsur–Levitzki type theorem for the Lie superalgebras \(\mathfrak{o}\mathfrak{s}\mathfrak{p}\left( {1,2n} \right)\)) inspired by Kostant's cohomological interpretation of the classical theorem. We show that the Lie superalgebras \(\mathfrak{g}\mathfrak{l}\left( {p,q} \right)\) cannot satisfy an Amitsur–Levitzki type super identity if pq≠0 and conjecture that neither can any other classical simple Lie superalgebra with the exception of \(\mathfrak{o}\mathfrak{s}\mathfrak{p}\left( {1,2n} \right)\).