**Workshop on homological devices and non-commutative algebra**

**May 8, 2007**** **

**Centre
for Mathematical Sciences, **

**Lund**** university**

**Sweden**** **

**Organized
by: **

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**,

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
Hellström**, Umeå:

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**

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,

Some Applications and Origins of N-complexes.

**Abstracts**

**Sergei**** Silvestrov**, ** **

**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.

**Lars Hellström**, Umeå university:

**Title:** A Generic Diamond Lemma with
applications for nonassociative algebra, operads,

Hopf algebras, and beyond

**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. Gröbner
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,

**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****, **

**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.