Last update: 17 September 2013
A finite dimensional complex semisimple Lie algebra is a finite dimensional Lie algebra over such that The following theorem classifies all finite dimensional complex semisimple Lie algebras.
|(a)||Every finite dimensional complex semisimple Lie algebra is a direct sum of complex simple Lie algebras.|
|(b)||There is one complex simple Lie algebra corresponding to each of the following types|
The complex simple Lie algebras of types and are the ones of classical type and they are where is the matrix of a skew-symmetric form on a space.
Let be a complex semisimple Lie algebra. A Cartan subalgebra of is a maximal abelian subalgebra of Fix a Cartan subalgebra of If is a finite dimensional and is any linear function, define The space is the space of It is a nontrivial theorem (see [Ser1987]) that where is a in which can be identified with the which is defined below in Appendix A8. The vector space is the space of linear functions from to
Let be the group algebra of It can be given explicitly as where the are formal variables indexed by the elements of The character of a is
Theorem (A7.1) is due to the founders of the theory, Cartan and Killing, from the late 1800’s. The beautiful text of Serre [Ser1987] gives a review of the definitions and theory of complex semisimple Lie algebras. See [Hum1978] for further details.
This is the survey paper Combinatorial Representation Theory, written by Hélène Barcelo and Arun Ram.
Key words and phrases. Algebraic combinatorics, representations.
Barcelo was supported in part by National Science Foundation grant DMS-9510655.
Ram was supported in part by National Science Foundation grant DMS-9622985.
This paper was written while both authors were in residence at MSRI. We are grateful for the hospitality and financial support of MSRI..