Combinatorial Representation Theory

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia
aram@unimelb.edu.au

Last update: 17 September 2013

Appendix A

A7. Complex semisimple Lie algebras

A finite dimensional complex semisimple Lie algebra is a finite dimensional Lie algebra $𝔤$ over $ℂ$ such that $\text{rad}\left(𝔤\right)=0\text{.}$ The following theorem classifies all finite dimensional complex semisimple Lie algebras.

Theorem A7.1

(a)	Every finite dimensional complex semisimple Lie algebra $𝔤$ is a direct sum of complex simple Lie algebras.
(b)	There is one complex simple Lie algebra corresponding to each of the following types $$A_{n-1},B_n,C_n,D_n,E_6,E_7,E_8,F_4,G_2\text{.}$$

The complex simple Lie algebras of types $A_n,$ $B_n,$ $C_n$ and $D_n$ are the ones of classical type and they are $$\begin{array}{cc}\text{Type}\hspace{0.17em}{A}_{n-1}\text{:}& 𝔰𝔩(n,ℂ)=\{A\in M_n\left(ℂ\right)\hspace{0.17em}|\hspace{0.17em}\text{Tr}\left(A\right)=0\},\\ \text{Type}\hspace{0.17em}{B}_n\text{:}& 𝔰𝔬(2n+1,ℂ)=\{A\in M_{2n+1}\left(ℂ\right)\hspace{0.17em}|\hspace{0.17em}A+A^t=0\},\\ \text{Type}\hspace{0.17em}{C}_n\text{:}& 𝔰𝔭(2n,ℂ)=\{A\in M_{2n}\left(ℂ\right)\hspace{0.17em}|\hspace{0.17em}AJ+J{A}^t=0\},\\ \text{Type}\hspace{0.17em}{D}_n\text{:}& 𝔰𝔬\left(2n\right)=\{A\in M_{2n}\left(ℂ\right)\hspace{0.17em}|\hspace{0.17em}A+A^t=0\},\end{array}$$ where $J$ is the matrix of a skew-symmetric form on a $2n\text{-dimensional}$ space.

Let $𝔤$ be a complex semisimple Lie algebra. A Cartan subalgebra of $𝔤$ is a maximal abelian subalgebra $𝔥$ of $𝔤\text{.}$ Fix a Cartan subalgebra $𝔥$ of $𝔤\text{.}$ If $V$ is a finite dimensional $𝔤\text{-module}$ and $\mu :𝔥\to ℂ$ is any linear function, define $$V_{\mu }=\{v\in V\hspace{0.17em}|\hspace{0.17em}hv=\mu \left(h\right)v,\hspace{0.17em}\text{for all}\hspace{0.17em}h\in 𝔥\}\text{.}$$ The space $V_{\mu }$ is the $\mu \text{-weight}$ space of $V\text{.}$ It is a nontrivial theorem (see [Ser1987]) that $$V=\underset{\mu \in P}{⨁}{V}_{\mu },$$ where $P$ is a $\mathbb{Z}\text{-lattice}$ in ${𝔥}^{*}$ which can be identified with the $\mathbb{Z}\text{-lattice}$ $P$ which is defined below in Appendix A8. The vector space ${𝔥}^{*}$ is the space of linear functions from $𝔥$ to $ℂ\text{.}$

Let $ℂ\left[P\right]$ be the group algebra of $P\text{.}$ It can be given explicitly as $$ℂ\left[P\right]=ℂ\text{-span}\left\{e^{\mu }\hspace{0.17em}\right|\hspace{0.17em}\mu \in P\},\text{with multiplication}\hspace{0.17em}{e}^{\mu }{e}^{\nu }=e^{\mu +\nu },\hspace{0.17em}\text{for}\hspace{0.17em}\mu ,\nu \in P,$$ where the $e^{\mu }$ are formal variables indexed by the elements of $P\text{.}$ The character of a $𝔤\text{-module}$ is $$\text{char}\left(V\right)=\sum _{\mu \in P}\text{dim}\left(V_{\mu }\right)e^{\mu }\text{.}$$

References

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.

Notes and references

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..

page history