Hebrew University
 
Amitsur Algebra Seminar
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Time: Thursday, Apr 25, 12:00-13:15
 
Place: math 209
 
Title: General Bilinear Forms and Applications
 
Speaker: Uriya First (HebrewU)
Abstract: 
We introduce the new notion of general bilinear forms and present
several of its applications. In particular, we give new proofs to
theorems of Saltman and Osborn.
 
In more detail: 
Let F be a field, and let V be a finite dimensional vector space.
There is a well-known one-to-one correspondence between the regular
(=nondegenerate) bilinear forms on V, considered up to multiplication
by an element of F*, and the set of F-anti-automorphisms of End(V).
This result is the foundation of the widely studied connection
between hermitian forms and involutions.
We present the new notion of general bilinear forms, and use it to
prove a generalization of the correspondence to modules over
arbitrary rings: If R is a ring (not necessarily commutative,
possibly without involution) and M is a finitely generated projective
R-module, then there is a one-to-one correspondence between the
regular general bilinear forms over M, considered up to similarity,
and the set of anti-automorphisms of End_R(M). In particular, we
obtain a new proof of the classical correspondence (which does not
use the Skolem-Noether theorem).
The generalized correspondence is used to obtain new short proofs of:
(1) A theorem of Saltman: An Azumaya algebra is Brauer equivalent to
its opposite iff it is Brauer equivanet to an algebra with an
involution of the first kind. (The notions will be defined and
explained in the lecture.)
(2) A result of Osborn classifying rings with involution in which all
elements invariant under the involution are invertible or nilpotet.
Both results are in fact slightly generalized.
Other applications of the correspondence will be presented if time
permits.
 
You are cordially invited!
 
 <http://math.huji.ac.il/amitsur.html>
 
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Technion Math Net-2 (TECHMATH2)
Editor: Michael Cwikel   <techm@math.technion.ac.il> 
Announcement from: Konstantin Golubev   <kgolubev@gmail.com>