Workshop on homological devices and non-commutative algebra
Centre for Mathematical Sciences,
Tuesday, May 8, 13.15-14:00 in MH:309B:
Twisted Derivations and Introduction to Quasi Lie Algebras.
Tuesday, May 8, 14.05-15:00 in MH:309B:
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:
N-Complexes and Precohomologies, [NCPC].
Tuesday, May 8, 16.30-17:30 in MH:309B:
Some Applications and Origins of N-complexes.
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
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.).
Larsson, Institute Mittag-Leffler,
Title: Some applications and origins of
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.
Title: N-Complexes and Precohomologies,
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.