On semi-simple Lie algebras

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

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...

Koszul duality and modular representations of semi-simple Lie algebras

In this paper we prove that if G is a connected, simplyconnected, semi-simple algebraic group over an algebraically closed field of sufficiently large characteristic, then all the blocks of the restricted enveloping algebra (Ug)0 of the Lie algebra g of G can be endowed with a Koszul grading (extending results of Andersen, Jantzen, Soergel). We also give information about the Koszul dual rings....

Quantization of Cohomology in Semi-simple Lie Algebras

Let g be a complex, finite-dimensional Lie algebra, and Mo a contractable neighborhood of a complex homogeneous space on which g acts transitively. The present article investigates quantization of H(g; Cω(Mo)/C) by the condition that the corresponding realization by first-order operators admits a finitedimensional invariant subspace of functions. The quantization of cohomology phenomenon surfac...