Hom-Lie algebra structures on semi-simple Lie algebras

Enveloping Algebras of Hom-lie Algebras

A Hom-Lie algebra is a triple (L, [−,−], α), where α is a linear self-map, in which the skew-symmetric bracket satisfies an α-twisted variant of the Jacobi identity, called the Hom-Jacobi identity. When α is the identity map, the Hom-Jacobi identity reduces to the usual Jacobi identity, and L is a Lie algebra. Hom-Lie algebras and related algebras were introduced in [1] to construct deformation...

Representations of Complex Semi-simple Lie Groups and Lie Algebras

This article is an exposition of the 1967 paper by Parthasarathy, Ranga Rao, and Varadarajan, on irreducible admissible Harish-Chandra modules over complex semisimple Lie groups and Lie algebras. It was written in Winter 2012 to be part of a special collection organized to mark 10 years and 25 volumes of the series Texts and Readings in Mathematics (TRIM). Each article in this collection is int...

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 ...

Semi-simple Lie Algebras and Their Representations

This paper presents an overview of the representations of Lie algebras, particularly semi-simple Lie algebras, with a view towards theoretical physics. We proceed from the relationship between Lie algebras and Lie groups to more abstract characterizations of Lie groups, give basic definitions of different properties that may characterize Lie groups, and then prove results about root systems, pr...

Semi-Simple Lie Algebras and Their Representations