Involutions and trace forms on exterior powers of a central simple algebra

Abstract

For $A$ a central simple algebra of degree $2n$, the $n$th exterior power algebra ${\lambda }^{n}A$ is endowed with an involution which provides an interesting invariant of $A$. In the case where $A$ is isomorphic to $Q\otimes B$ for some quaternion algebra $Q$, we describe this involution quite explicitly in terms of the norm form for $Q$ and the corresponding involution for $B$.