Event № 317
Event № 317
TE Algebra Seminar - Uriya First 17/01/2013, Thursday, 13:30
Name: Algebra Seminar
Title: Non-Reflexive Hermitian Categories and Systems of Bilinear Forms.
Speaker: Uriya First
Place: Amado 814, Technion
A hermitian category is a triplet (H,*,w) such that H is an additive category, * is contravariant functor from H to itself and w: id --> ** is a natural isomorphism satisfying a certain identity (e.g. take H to be the f.d. vector spaces and * to be the functor sending V to V*). Heremitian categories are categorical frameworks for quadratic and bilinear forms that allow to prove results about them in great generality. Let (H,*,w) be a hermitian category. I call H non-reflexive if w : id --> ** is only assumed to be a natural transformation, rather than a natural isomorphism. Most results about hermitian categories only apply to the reflexive case (i.e. when w is an isomorphism). In this talk I will show that given a non-reflexive category (H,*,w), there exists a reflexive category (H',*',w') such that the category of arbitrary bilinear forms over (H,*,w) (even non-symmetric forms) is equivalent to the category of symmetric regular (=unimodular) bilinear forms over (H',*',w'). Next, I will show how systems of bilinear forms can be understood as a single bilinear form in an appropriate non-reflexive hermitian category. Combining both observations leads to numerous applications to systems of bilinear forms and also to hermitian forms over rings which are defined over non-reflexive modules. Among these applications are Witt's Cancellation Theorem, a version of Springer's Theorem, and various results about isometry of (systems of) bilinear forms.
SubmittedBy: u. elias , firstname.lastname@example.org
EventLink: Event № 317