Jorge Morales’s research while affiliated with Louisiana State University and other places

Let k be a field of characteristic different from 2 and let T be a fixed k-torus of dimension n. In this paper we study faithful k-representations ρ:T→SO(A,σ), where (A,σ) is a central simple algebra of degree 2n with orthogonal involution σ. Note that in this case ρ(T) is a maximal torus in SO(A,σ). We are interested in describing the pairs (A,σ) for which there is such a representation. We compute invariants for these algebras (discriminant and Clifford algebra), which are sufficient to determine their isomorphism class when I3(k)=0 by a theorem of Lewis and Tignol. The first part of the paper is devoted to the case where A is split over k and an application to a theorem of Feit on orthogonal groups over Q is given.

 Let A be a central simple algebra of degree n over a field of characteristic different from 2 and let B ? A be a maximal commutative subalgebra. We show that if there is an involution on A that preserves B and such that the socle of each local component of B is a homogeneous C 2 -module for this action, then B is a Frobenius algebra. For a fixed commutative Frobenius algebra B of finite dimension n equipped with an involution σ, we characterize the central simple algebras A of degree n that contain B and carry involutions extending σ.

The second Stiefel–Whitney class of the quadratic form trA/k(ax2) is computed, where A is a central simple algebra over a perfect field k of characteristic different from 2, a [set membership] A is a fixed element, and trA/k is the reduced trace. This class is related on the one hand to the class of A in the Brauer group, and on the other hand to corestrictions of quaternion algebras over certain factors arising from E[otimes B: multiply sign in circle]kE, where E is a commutative étale algebra over k that depends on the semisimple part of a.

Let Λ be a ring endowed with an involution a → ã. We say that two units a and b of Λ fixed under the involution are congruent if there exists an element u ∈ Λx such that a = ubũ. We denote by H(Λ) the set of congruence classes. In this paper we consider the case where Λ is an order with involution in a semisimple algebra A over a local field and study the question of whether the natural map H(Λ) → H(A) induced by inclusion is injective. We give sufficient conditions on the order Λ for this map to be injective and give applications to hermitian forms over group rings.

Nous montrons que l’invariant de Hasse-Witt de la forme de Killing d’une algèbre de Lie semi-simple L s’exprime à l’aide de l’invariant de Tits de la représentation irréductible de L de poids dominant ρ = 1/2 (somme des racines positives), et des invariants associés au groupe des symétries du diagramme de Dynkin de L.