First cohomology groups of soluble lie algebras

Cohomology of Lie groups and Lie algebras

The aim of this expository essay is to illustrate one example of a local-to-global phenomenon. What we mean by that is better illustrated by explaining the topic at hand. We aim to understand the de Rham cohomology groups of a Lie group. But instead of doing it using actual differential forms, we shall use the properties of a Lie group (especially the fact that it is a group) to reduce calculat...

Semiinfinite Cohomology of Lie-* Algebras

This paper is a natural extension of the previous note [Ar2]. Semiinfinite cohomology of Tate Lie algebra was defined in that note in terms of some duality resembling Koszul duality. The language of differential graded Lie algebroids was the main technical tool of the note. The present note is devoted to globalization of the main construction from [Ar2] in the following sense. The setup in [Ar2...

Cohomology of Restricted Lie Algebras

In this dissertation, we investigate the cohomology theory of restricted Lie algebras. Motivations for the definition of a restricted Lie algebra are given and the theory of ordinary Lie algebra cohomology is briefly reviewed, including a discussion on algebraic interpretations of the low dimensional cohomology spaces of ordinary Lie algebras. The general Cartan-Eilenberg construction of the st...

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...

Lie Groups and Lie Algebras