Workshop on homological devices and non-commutative algebra

 

May 8, 2007

 

Centre for Mathematical Sciences,

Lund university

Sweden

 

Organized by:

 

Sergei Silvestrov  

sergei.silvestrov@math.lth.se

http://www.maths.lth.se/matematiklth/personal/ssilvest/

http://www.maths.lth.se/matematiklth/personal/ssilvest/nangseminar.html

 

Supported by:

 

Crafoord Foundation and STINT Foundation

 

Program

 

Tuesday, May 8, 13.15-14:00 in MH:309B:

Noncommutative geometry och noncommutative analysis seminar

Sergei Silvestrov, Lund:

Twisted Derivations and Introduction to Quasi Lie Algebras.
 

Tuesday, May 8, 14.05-15:00 in MH:309B:

Noncommutative geometry och noncommutative analysis seminar

Lars Hellstr÷m, Umeň:

A Generic Diamond Lemma with Applications for Nonassociative Algebra, Operads, Hopf Algebras, and Beyond.
 

Tuesday, May 8, kl 15.30-16:30 in MH:309B:

Noncommutative geometry och noncommutative analysis seminar

Goro Kato, California Polytechnic State University, San Luis Obispo:

N-Complexes and Precohomologies, [NCPC].
 

Tuesday, May 8, 16.30-17:30 in MH:309B:

Noncommutative geometry och noncommutative analysis seminar

Daniel Larsson, Institute Mittag-Leffler, Stockholm:

Some Applications and Origins of N-complexes.

 

Abstracts

 

Sergei Silvestrov, Lund University

Title: Twisted Derivations and Introduction to Quasi Lie Algebras.

Abstract: In this talk, I will present a review on quasi-Lie algebras, quas-hom-Lie algebrasand hom-Lie algebras generalizing color Lie algebras, Lie superalgebras and Lie algebras and providing a unifying framework for quasi-Lie quasi-deformations via twisted derivations of infinite-dimensional Lie algebras of vector fields of Witt and Virasoro type.á

 

Lars Hellstr÷m, Umeň university:

Title: A Generic Diamond Lemma with applications for nonassociative algebra, operads,
Hopf algebras, and beyond
á
Abstract: When studying algebraic structures defined by generators and relations, one often relies on a diamond lemma (or some more specialised counterpart, such as may be found in e.g. Gr÷bner basis theory) to obtain an effective model for the structure studied. A problem is however that diamond lemmas tend to be stated only for a specific family of algebraic structures, so that each new kind of algebraic structures (with different axioms, a different set of operations, or whatever) requires its own diamond lemma. In this talk, I will present my Generic Diamond Lemma and explain how one can apply it for a large variety of algebraic structures. Focus will be on (i) how a single result can handle many different kinds of multiplicative structures (commutative, associative, free non-associative, path algebra, etc.) and (ii) how to handle many-sorted algebraic structures (operads, PROPs, symocats, etc.).

 

Daniel Larsson, Institute Mittag-Leffler, Stockholm:

Title: Some applications and origins of N-complexes

Abstract: In this talk I will discuss definitions, motivations and applications, present and future, of so-called N-complexes. These generalize ordinary complexes in homological algebra in that one do not assume the differentials to be square-zero. Instead they are nilpotent of some higher order.

 

Goro Kato, California Polytechnic State University, San Luis Obispo,

Title: N-Complexes and Precohomologies, [NCPC]

Abstract: For an abelian category A, we can from the category Se(A) of  sequences and morphisms of A.  Recall that in order to define  cohomologies we need the category of complexes.  There is a new  invariant, generalizing the notion of a cohomology, defined from Se (A) to A.  This invariant is said to be a precohomology.  There exist  two functors from Se(A) to the category of N-complexes, where N>1. Those functors are said to be complexifying functors.  Then we can  define a cohomology on the complexified objects in the category of N-complexes, which is the precohomology of (i,k)-type.  In order to define the Derived Category on N-complexes, we need several projects  including the notions of quasi-isomorphism of N-complexes, homotopy  of N-complexes, spectral sequences based on N-complexes and  precohomologies of composite functors, hyperprecohomologies.  Also one can define a notion of inverse limits connecting to  precohomologies
We will outline those projects and main statements.