We prove an Amitsur–Levitzki type theorem for the Lie superalgebras
\mathfrako\mathfraks\mathfrakp( 1,2n )\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
\mathfrakg\mathfrakl( p,q )\mathfrak{g}\mathfrak{l}\left( {p,q} \right)
cannot satisfy an Amitsur–Levitzki type super identity if
pq
\mathfrako\mathfraks\mathfrakp( 1,2n )\mathfrak{o}\mathfrak{s}\mathfrak{p}\left( {1,2n} \right)
.
Amitsur–Levitzki theorem - Lie superalgebras - transgression operator