HAL CCSD
DEGREE 3 RELATIVE INVARIANT FOR UNITARY INVOLUTIONS
Barry, Demba
Masquelein, Alexandre
Quéguiner-Mathieu, Anne
Université des Sciences, des Techniques et des Technologies de Bamako (USTTB)
Laboratoire Analyse, Géométrie et Applications (LAGA) ; Université Paris 8 Vincennes-Saint-Denis (UP8)-Université Paris 13 (UP13)-Institut Galilée-Centre National de la Recherche Scientifique (CNRS)
hal-03668652
https://hal.science/hal-03668652
https://hal.science/hal-03668652v1/document
https://hal.science/hal-03668652v1/file/BMQTFinal.pdf
https://hal.science/hal-03668652
2022
en
[MATH]Mathematics [math]
[MATH.MATH-RA]Mathematics [math]/Rings and Algebras [math.RA]
info:eu-repo/semantics/preprint
Preprints, Working Papers, ...
Using the Rost invariant for non split simply connected groups, we define a relative degree 3 cohomological invariant for pairs of orthogonal or unitary involutions having isomorphic Clifford or discriminant algebras. The main purpose of this paper is to study general properties of this invariant in the unitary case, that is for torsors under groups of outer type A. If the underlying algebra is split, it can be reinterpreted in terms of the Arason invariant of quadratic forms, using the trace form of a hermitian form. When the algebra with unitary involution has a symplectic or orthogonal descent, or a symplectic or orthogonal quadratic extension, we provide comparison theorems between the corresponding invariants of unitary and orthogonal or symplectic types. We also prove the relative invariant is classifying in degree 4, at least up to conjugation by the non-trivial automorphism of the underlying quadratic extension. In general, choosing a particular base point, the relative invariant also produces absolute Arason invariants, under some additional condition on the underlying algebra. Notably, if the algebra has even co-index, so that it admits a hyperbolic involution, which is unique up to isomorphism, we get a so-called hyperbolic Arason invariant. Assuming in addition the algebra has degree 8, we may also define a decomposable Arason invariant. It generally does not coincide with the hyperbolic Arason invariant, as the hyperbolic involution need not be totally decomposable.
AFRIMath
2022-05-15
info:eu-repo/semantics/OpenAccess