For various series of complex semi-simple Lie algebras g(t) equipped with irreducible representations V (t), we decompose the tensor powers of V (t) into irreducible factors in a uniform manner, using a tool we call diagram induction. In particular, we interpret the decompostion formulas of Deligne [4] and Vogel [23] for decomposing g respectively for the exceptional series and k ≤ 4 and all si...

We study rigidity questions for pairs of Lie algebras [Formula: see text] admitting a post-Lie algebra structure. show that if is semisimple and arbitrary, then we have in the sense must be isomorphic. The proof uses result on decomposition as direct vector space sum two subalgebras. hence isomorphic to text]. This solves some open existence structures prove additional results text], where comp...

The classification of 2-step nilpotent Lie algebras is attacked by a generator-relation method. The main results are in low dimensions or a small number of relations. Introduction. According to a theorem of Levi, in characteristic zero a finitedimensional Lie algebra can be written as the direct sum of a semisimple subalgebra and its unique maximal solvable ideal. If the field is algebraically ...

Let g be a nonabelian Lie algebra over an algebraically closed field K of characteristic 0. One is interested in the (algebraically) irreducible representations of g acting on a vector space which is allowed to be infinite dimensional. The subject of enveloping algebras is largely concerned with these, but even in the simplest nonabelian case, with g = I) the 3-dimensional (nilpotent) Heisenber...

Let G be a quasi-split connected reductive group over a local field of characteristic 0, and fix a regular nilpotent element in the Lie algebra g of G. A theorem of Kostant then provides a canonical conjugacy class within each stable conjugacy class of regular semisimple elements in g. Normalized transfer factors take the value 1 on these canonical conjugacy classes.

We classify the left-invariant sub-Riemannian structures on unique five-dimensional simply connected two-step nilpotent Lie group with two-dimensional commutator subgroup; this 5D is first twostep beyond three-and Heisenberg groups. Alongside, we also present a classification, up to automorphism, of subspaces associated algebra (together complete set invariants).

A finite W-algebra U (g, e) is a certain finitely generated algebra that can be viewed as the enveloping algebra of the Slodowy slice to the adjoint orbit of a nilpotent element e of a complex reductive Lie algebra g. It is possible to give the tensor product of a U (g, e)-module with a finite dimensional U (g)-module the structure of a U (g, e)-module; we refer to such tensor products as trans...

Suppose that a finite group G admits a Frobenius group of automorphisms FH with kernel F and complement H such that the fixed-point subgroup of F is trivial: CG(F ) = 1. In this situation various properties of G are shown to be close to the corresponding properties of CG(H). By using Clifford’s theorem it is proved that the order |G| is bounded in terms of |H| and |CG(H)|, the rank of G is boun...

Let G be a complex simple algebraic group and let g be its Lie algebra. A nilpotent orbit O in g is an orbit of a nilpotent element of g by the adjoint action of G on g. Then O admits a natural symplectic 2-form ω and the nilpotent orbit closure Ō has symplectic singularities in the sense of [Be] and [Na3] (cf. [Pa], [Hi]). In [Ri], Richardson introduced the notion of so-called the Richadson or...

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...