Tag - Central simple algebras

Susanne Pumpluen: A way to generalize classical results from central simple algebras to the nonassociative setting

11th December, 2023

Recently, the theory of semiassociative algebras and their Brauer monoid was introduced by Blachar, Haile, Matri, Rein, and Vishne as a canonical generalization of the theory of associative central simple algebras and their Brauer group: together with the tensor product semiassociative algebras over a field form a monoid that contains the classical Brauer group as its unique maximal subgroup. We present classes of semiassociative algebras that are canonical generalizations of classes of certain central simple algebras and explore their behavior in the Brauer monoid. Time permitting, we also discuss some - hopefully interesting - particularities of this newly defined Brauer monoid.

Alberto Elduque: Graded-simple algebras and twisted loop algebras

8th October, 2020

The loop algebra construction by Allison, Berman, Faulkner, and Pianzola (2008), describes graded-central-simple algebras with "split centroid" in terms of central simple algebras graded by a quotient of the original grading group. Particular versions of this result were considered by several authors.

In this talk it will be shown how the restriction on the centroid can be removed, at the expense of allowing some deformations (cocycle twists) of the loop algebras.

Julia Hartmann: Local-Global Principles for Galois Cohomology

13th December, 2012

We consider Galois cohomology groups over function fields F of curves that are defined over a complete discretely valued field. Motivated by work of Kato and others for n=3, we show that local-global principles hold for Hn(F,ℤ/mℤ(n−1)) for all n>1. In the case n=1, a local-global principle need not hold. Instead, we will see that the obstruction to a local-global principle for H1(F,G), a Tate-Shafarevich set, can be described explicitly for many (not necessarily abelian) linear algebraic groups G. Concrete applications of the results include central simple algebras and Albert algebras.