Lambda-operations for hermitian forms over algebras with involution

Abstract

The Grothendieck-Witt ring of a field is known to be a $\lambda$-ring, where the $\lambda$-operations are induced by the exterior powers of bilinear spaces. We give a similar construction on the mixed Grothendieck-Witt ring of a central simple algebra with involution over a field. In doing so we also develop a general framework for pre-$\lambda$-ring structures on semi-rings graded over a monoid. When the involution is of the first kind, the construction is improved, and even $\lambda$-powers of a hermitian form are quadratic forms, while odd $\lambda$-powers are hermitian forms over the same algebra. Some explicit computations of those spaces are given, as well as connexions with restrictions of trace forms. We also explain how to apply this theory to define the determinant of an involution in a larger setting than was previously done.