FILE:liealgeb.txt

ITEM: Lie Algebras, Version: 1

Mathematica version: 4.0.1.0, for Linux 

DATE: November 22, 1999

AUTHOR: Rolf Sulanke

ADDRESS:
sulanke@mathematik.hu-berlin.de

Homepage:
http://www-irm.mathematik.hu-berlin.de/~sulanke/
          
CONTENTS:
liealgeb.nb
Declare.m
liealg.m
liealgun.m
liealgeb.txt

KEYWORDS:
indefinite scalar products, Lie algebra, semisimple Lie algebra, compact Lie 
algebra, orthogonal Lie algebra, pseudo-orthogonal Lie algebra, linear Lie 
algebra, special linear Lie algebra, symplectic Lie algebra, unitary Lie 
algebra, special unitary Lie algebra, orthogonalization,  Killing form, 
commutator, Jacoby identity, structure constants, center. 

SUMMARY:
THE NOTEBOOK:
In the notebook liealgeb.nb we define basic operations for Lie algebras of 
matrices, and calculate the Killing forms of the following Lie algebras:
the general linear Lie algebras gl(n,K),
the special linear  Lie algebras sl(n,K),
the orthogonal Lie algebras o(n), 
the pseudo-orthogonal Lie algebras o(n, l),
the symplectic Lie algebras sp(2n, K),
the unitary Lie algebras u(n),
the special unitary Lie algebras su(n).
Here K denotes the fields of the real or the complex numbers. 
Furthermore we construct an orthonormalization procedure appropriate for 
arbitrary symmetric  bilinear forms defined on spaces of matrices, in 
particular for Killing forms. By the help of this orthonormalization we 
calculate the index of the considered Killing forms (in the real case, for 
small values of n). In principle the tools developed in this notebook give the 
possibility to perform any  calculation in the series of Lie algebras listed 
above. 

THE PACKAGES:
The package liealg.m contains the essential constructs developed in the 
notebook liealgeb.nb, with exception of those concerning the unitary and 
special unitary Lie algebras, which are collected in the package liealgun.m.
The reason for separating the unitary case is that these Lie algebras are real 
Lie algebras, in spite of the fact that their elements are matrices with 
complex elements. Thus the coordinates for the elements of the unitary Lie 
algebras must be declared to be real, what we did by the help of the package

PEKKA JANHUNEN: Declare.m,  in 
http://www.mathsource.com/Content22/Enhancements/Algebraic/0202-149. 

For the convenience of the user, and since we added a comment and changed the 
Context, this package has been included in the item Lie Algebras.

GENERAL HINTS:

1. Put all the submitted files  into the same directory, and
you will not suffer from Path problems.

2. All private (not delivered from the Mathematica system) symbols defined in
the submitted files (with eception of Declare.m)) start with small letters.

3. The notebook liealgeb.nb needs importing Declare.m only for section 6; do 
not import the other packages. Since the symbols in the packages are protected 
you will get error messages if you run the notebook having imported the 
packages. Use the notebook for testing puposes, and if you want to modify the 
definitions, or to include other Lie algebras.
 
4. For using the concepts defined in the packages it suffices to import  
liealgun.m, which needs liealg.m and Declare.m, and imports them 
automatically. If you want to consider only the non unitary Lie algebras it 
suffices to import liealg.m

5. The notebooks and packages in the collection "Lie Algebras" are free 
software. Any user may change and adapt them to his aims. If publishing such 
adaptions or applications, please cite the sources and sign with your name as 
the author. I am grateful for copies of such applications, for your hints,
corrections, comments etc. Please, e-mail them to the address above or to

rolf.sulanke@t-online.de

6. Please, excuse posible errors and bad, too simplified use of anglo-american 
English in my texts. I never learnt the English systematically.