瑞典Mälardalen大学应用数学系教授、Scientific leader for Mathematics and Applied Mathematics Research Environment (MAM)
In this talk, an introductory overview will be presented of some recent developments on the subject of Hom-algebra, with focus on quasi Lie algebras, quasi-Hom Lie algebras, Hom-Lie algebras and related Hom-algebra structures. These objects appear for example when discretizing the differential calculus. Quasi Lie algebras encompass in a natural way the Lie algebras, Lie superalgebras, color Lie algebras, Hom-Lie algebras, q-Lie algebras (proposed by Prof. N.H. Hu) and various algebras of discrete and twisted vector fields arising for example in connection to algebras of twisted discretized derivations, Ore extension algebras, q-deformed vertex operators structures and q-deferential calculus, multiparameter deformations of associative and non-associative algebras, 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 choices of parameters algebras appearing in non-commutative algebraic geometry, universal enveloping algebras of Lie algebras, Lie superalgebras and color Lie algebras and their deformations. Common unifying feature for all these algebras is appearance of some twisted generalizations of Jacoby identities providing new structures of interest for investigation from the side of associative algebras, non-associative algebras, generalizations of Hopf algebras, non-commutative differential calculi beyond usual differential calculus and generalized quasi-Lie algebra central extensions and Hom-algebra formal deformations and co-homology. Also in the talk, I will describe some related n-ary Hom-algebra generalizations of Nambu algebras, associative algebras and Lie algebras.