We give a comprehensive survey of the theory of finite dimensional Lie algebras over an algebraically closed field of prime characteristic and announce that the classification of all finite dimensional simple Lie algebras over an algebraically closed field of characteristic p > 3 is now complete. Any such Lie algebra is up to isomorphism either classical or a filtered Lie algebra of Cartan type...

In contrast with the finite dimensional case, one of the distinguished features in the theory of infinite dimensional Lie algebras is the modular invariance of the characters of certain representations. It is known [Fr], [KP] that for a given affine Lie algebra, the linear space spanned by the characters of the integrable highest weight modules with a fixed level is invariant under the usual ac...

Small print: These notes are intended to give the logical structure of the first half of the course; proofs and further remarks will be given in lectures. I would very much appreciate being told of any corrections or possible improvements. These notes, together with some brief notes on the linear algebra prerequisites, and the problem sheets for the course, are available via my home page, http:...

The notion of Poisson manifold with compatible pseudo-metric was introduced by the author in [1]. In this paper, we introduce a new class of Lie algebras which we call a pseudo-Riemannian Lie algebras. The two notions are strongly related: we prove that a linear Poisson structure on the dual of a Lie algebra has a compatible pseudo-metric if and only if the Lie algebra is a pseudo-Riemannian Li...

Introduction For this course you need a very good understanding of linear algebra; a good knowledge of group theory and the representation theory of finite groups will also help. The main sources for these notes are the books [6] and [8]. We give complete proofs of all statements with the exception of the conjugacy of Cartan subgroups, the uniqueness theorem for semisimple Lie algebras, and the...

An algebraic algorithm is developed for computation of invariants (‘generalized Casimir operators’) of general Lie algebras over the real or complex number field. Its main tools are the Cartan’s method of moving frames and the knowledge of the group of inner automorphisms of each Lie algebra. Unlike the first application of the algorithm in [J.Phys. A: Math. Gen., 2006, V.39, 5749; math-ph/0602...

We find a natural compatible condition between the Rota-Baxter operator and Turaev's (Hopf) group-(co)algebras, which leads to concept of group-(co)algebra. Two characterizations group-algebras (abbr. T-algebras) are obtained: one by Atkinson factorization other T-quasi-idempotent elements. The relations among some related group algebraic structures (such as (tri)dendriform T-algebras, Zinbiel ...

We prove that every Lie algebra can be decomposed into a solvable Lie algebra and a semisimple Lie algebra. Then we show that every complex semisimple Lie algebra is a direct sum of simple Lie algebras. Finally, we give a complete classification of simple complex Lie algebras.