Double affine Lie algebras and finite groups

Double Affine Lie Algebras and Finite Groups

We begin to study the Lie theoretical analogs of symplectic reflection algebras for Γ a finite cyclic group, which we call “cyclic double affine Lie algebra”. We focus on type A : in the finite (resp. affine, double affine) case, we prove that these structures are finite (resp. affine, toroidal) type Lie algebras, but the gradings differ. The case which is essentially new is sln(C[u, v] o Γ). W...

Lie Algebras of Finite and Affine Type

Lie $^*$-double derivations on Lie $C^*$-algebras

A unital $C^*$ -- algebra $mathcal A,$ endowed withthe Lie product $[x,y]=xy- yx$ on $mathcal A,$ is called a Lie$C^*$ -- algebra. Let $mathcal A$ be a Lie $C^*$ -- algebra and$g,h:mathcal A to mathcal A$ be $Bbb C$ -- linear mappings. A$Bbb C$ -- linear mapping $f:mathcal A to mathcal A$ is calleda Lie $(g,h)$ -- double derivation if$f([a,b])=[f(a),b]+[a,f(b)]+[g(a),h(b)]+[h(a),g(b)]$ for all ...

Note on Lie algebras , finite groups and finite geometries

The subject of this note began with Thompson [Thl,2]. In the course of constructing his simple group Th, he considered the Lie algebra Lover iC of type Es. He constructed a decomposition L H1 ..L ... ..L H31 using a family H. {H1, ... , H31 } of Cartan algebras that are pairwise perpendicular with respect to the Killing form (he called this a "Dempwolff decomposition" of L; the construction was...

Lie Algebras, Algebraic Groups, and Lie Groups

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...