Sergei Silvestrov: Quasi-Lie algebras, quasi-Lie central extensions and quasi-deformations of Lie and color Lie algebras.

Abstract: In this talk the class of quasi-Lie algebras and some of its subclasses will be presented, including general construction methods and examples coming from twisted derivations and twisted generalized vector fields, color Lie algebras, generalized quasi-Lie central extensions of Lie algebras and superalgebras. Among examples arising within quasi-Lie algebras framework are known and new one-parameter and multi-parameter deformations of infinite-dimensional Lie algebras of Witt and Virasoro type some of which appear in the context of conformal field theory, string theory and deformed vertex models, multi-parameter families of quadratic and almost quadratic algebras that include for special natural choices of parameters algebras appearing in non-commutative algebraic geometry, as well as universal enveloping algebras of Lie algebras, Lie superalgebras and color Lie algebras. Common uniting feature for all these algebras is appearance of twisted generalizations of Jacoby identities providing new structures of interest for investigation from the side of associative algebras, the non-associative algebras, twisted generalizations of Hopf algebras, the non-commutative differential calculi beyond usual differential calculus and generalized central and other (co-)homological extensions.

[1] D. Larsson, S. Silvestrov, Quasi-Lie algebras, in "Noncommutative Geometry and Representation Theory in Mathematical Physics", AMS, Contemporary Mathematics, Vol. 391, 2005, 7 pp .

[2] J. Hartwig, D. Larsson, S. Silvestrov, Deformations of Lie algebras using sigma-derivations, J. Algebra 295 (2006), no. 2, 314-361.

[3] D. Larsson, S. Silvestrov, Quasi-hom-Lie Algebras, Central Extensions and 2-cocycle-like Identities, J. Algebra 288 (2005), no. 2, 321-344.

[4] D. Larsson, S. Silvestrov, Quasi-deformations of sl(2,K) using twisted derivations, Communications in Algebra, 35, 2007, 4303-4318.