FILE:liealg.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 liealgeb.nb:
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