HALF-LIE GROUPS

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...

SEMINAR ON LIE GROUPS 1. Lie Groups

Example 1.3. (R,+) Example 1.4. S or T n = S × ...× S Example 1.5. Gl (n,F) ⊆ F, where F = R or C Example 1.6. E3 = isometries of R (2 connected components) Let the orthogonal group O3 < E3 be the subgroup that fixes the origin, and let the special orthogonal group SO (3) = SO3 < O3 be the orientation-preserving elements of O3. Visualizing SO (3): Let u be a vector of length l in R, correspondi...

The Half-Told Lie

M y half-told lie is the claim that Melissa chose me as her research advisor. But with only a modest effort, you could paw through the thousands of emails on my server and find the one in which I recruited her to pursue a project on civilizing the Internet. Yet as it is only half a lie, the half-truth is the fact that she appeared on my doorstep unannounced two years ago and asked how she might...

Lie Algebras and Lie Brackets of Lie Groups–matrix Groups

The goal of this paper is to study Lie groups, specifically matrix groups. We will begin by introducing two examples: GLn(R) and SLn(R). Then in each section we will prove basic results about our two examples and then generalize these results to general matrix groups.

Lie Groups and Lie Algebras