We continue a previous study on Γ-vertex algebras and their quasimodules. In this paper we refine certain known results and we prove that for any Z-graded vertex algebra V and a positive integer N , the category of V -modules is naturally isomorphic to the category of quasimodules of a certain type for V ⊗N . We also study certain generalizations of twisted affine Lie algebras and we relate suc...

The structure constants of quantum Lie algebras depend on a quantum deformation parameter q and they reduce to the classical structure constants of a Lie algebra at q = 1. We explain the relationship between the structure constants of quantum Lie algebras and quantum Clebsch-Gordan coefficients for adjoint ⊗ adjoint → adjoint. We present a practical method for the determination of these quantum...

The problem of providing complete presentations reduction algebras associated to a pair Lie $$( \mathfrak{G} , \mathfrak{g} )$$ has previously been considered by Khoroshkin and Ogievetsky in the case diagonal algebra for $$\mathfrak{gl}(n)$$ . In this paper, we consider superalgebras \mathfrak{osp}(1|2) \times as double coset space having an associative $$ \mathrel{\scriptstyle\lozenge} -produc...

We describe an algorithm for computing automorphism groups and testing isomorphisms of finite dimensional Lie algebras over finite fields. The algorithm is particularly effective for simple or almost simple Lie algebras. We show how it can be used in a computer search for new small-dimensional simple Lie algebras over the field with two elements.

We consider the class of complex Lie algebras for which the Koszul 3-form is zero, and prove that it contains all quotients of Borel subalgebras, or of their nilradicals, of finite dimensional complex semisimple Lie algebras. A list of Kac-Moody types for indecomposable nilpotent complex Lie algebras of dimension ≤ 7 is given.

Generalizing Feingold-Frenkel’s construction we use Weyl bosonic fields to construct toroidal Lie algebras of types An, Bn, Cn and Dn of level −1,−2,−1/2 and −2 respectively. In particular, our construction also gives new bosonic construction for the orthogonal Lie algebras in the cases of affine Lie algebras.

These notes are an introduction to Lie algebras, algebraic groups, and Lie groups in characteristic zero, emphasizing the relationships between these objects visible in their categories of representations. Eventually these notes will consist of three chapters, each about 100 pages long, and a short appendix. Single paper copies for noncommercial personal use may be made without explicit permiss...