Quasi-Lie Algebras and Quasi-Deformations. Algebraic Structures Associated with Twisted Derivations

Abstract: This thesis introduces a new deformation scheme for Lie algebras, which we refer to as ?quasi-deformations? to clearly distinguish it from the classical Grothendieck-Schlessinger and Gers-tenhaber deformation schemes. The main difference is that quasi-deformations are not in gene-ral category-preserving, i.e., quasi-deforming a Lie algebra gives an object in the larger catego-ry of ?quasi-Lie algebras?, a notion which is also introduced in this thesis. The quasi-deforma-tion scheme can be loosely described as follows: represent a Lie algebra by derivations acting on a commutative, associative algebra with unity and replace these derivations with twisted versions. An algebra structure is then imposed, thus arriving at the quasi-deformed algebra. Therefore the quasi-deformation takes place on the level of representations: we ?deform? the representation, which is then ?pulled-back? to an algebra structure. The different Chapters of this thesis is concerned with different aspects of this quasi-deforma-tion scheme, for instance: Burchnall-Chaundy theory for the q-deformed Heisenberg algebra (Chapter II), the (quasi-Lie) algebraic structure on the vector space of twisted derivations (Chapter III), deformed Witt, Virasoro and loop algebras (Chapter III and IV), Central exten-sion theory (Chapter III and IV), the Lie algebra sl(2) and some associated quadratic algebras.