Shortest ever Intro to Lie Algebras and Lie Groups

A Lie algebra is simply

  • a) a vector space
  • b) with an “operation” such that the operation [x,y] of any two vectors x and y is again a vector, and such that the following hold:
  1. skew-symmetry: [x,y] = -[y,x].
  2. Bi-linearity:    [x,ay] = a[x,y],  [x,y+z] = [x,y] + [x,z].  (a is a number.)                               
  3. Jacobi identity: [x,[y,z]] + [y,[z,x]] + [z,[x,y]] = 0.

The best known example is the Vector Space R3 with the cross product as the operation.

The real importance of Lie algebras is that one can get one from any Lie group – roughly speaking, a group that’s also a manifold

Lie groups  crop up as the PRIMARY groups of SYMMETRIES in physics. The Lie algebra is essentially the infinitesimal version of the corresponding Lie group.

E.g. – The relation between the group of rotations in R3 and the cross product.

Here the group is called SO(3) and the Lie algebra is called so(3).

(So R^3 with its cross product is called so(3).) One can generalize this to any number of dimensions, letting SO(n) denote the group of rotations in R^n and so(n) the corresponding Lie algebra. (However, so(n) is not isomorphic to R^n except for n = 3, so there is something very special about three dimensions.)

Specializing in high volume web and cloud application architecture, Anuj Varma’s customer base includes Fortune 100 companies (dell.com, British Petroleum, Schlumberger).
Anuj’s training as a mathematical physicist followed by years of advanced computer programming is unique in the industry.

For Anuj’s popular technology seminars and science and scientific computing seminars, please visit ANUJ.COM

For Anuj’s Mathematical Models and Math Modeling related consulting , please visit anuj.com.

All content on this site is original and owned by AdverSite Web Holdings, Inc. – the parent company of anujvarma.com. No part of it may be reproduced without EXPLICIT consent from the owner of the content.

Anuj Varma – who has written posts on Anuj Varma, Technology Architect.


Leave a Reply

Your email address will not be published. Required fields are marked *