Simple Lie algebra

Note: This draft, when it becomes matured and informative, will replace the current redirect simple Lie algebra (see Talk:Simple_Lie_algebra for the reason why this page should start as a draft.)

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In algebra, a simple Lie algebra is a nonabelian Lie algebra contains no nonzero proper ideals. The classification of real simple Lie algebras is one of major achievements of Wilhelm Killing and Élie Cartan.

Complex simple Lie algebras

A finite-dimensional simple complex Lie algebra is isomorphic to either of the following: ${\mathfrak {sl}}_{n}\mathbb {C}$, ${\mathfrak {so}}_{n}\mathbb {C}$, ${\mathfrak {sp}}_{2n}\mathbb {C}$ (classical Lie algebras) or one of the five exceptional Lie algebras.

To each finite-dimensional complex semisimple Lie algebra ${\mathfrak {g}}$, there exists a corresponding diagram (called the Dynkin diagram) where the nodes denote the simple roots and the nodes are jointed (or not jointed) by a number of lines depending on the angles between the simple roots. The Dynkin diagram of ${\mathfrak {g}}$ is connected if and only if ${\mathfrak {g}}$ is simple. All possible connected Dynkin diagrams are the following:

The correspondence of the diagrams and complex simple Lie algebras are:

(A_n) $\quad {\mathfrak {sl}}_{n}\mathbb {C}$

(B_n) $\quad {\mathfrak {so}}_{2n+1}\mathbb {C}$

(C_n) $\quad {\mathfrak {sp}}_{2n}\mathbb {C}$

(D_n) $\quad {\mathfrak {so}}_{2n}\mathbb {C}$

See also

• Simple Lie group
• Satake diagram

References

• Fulton, William; Harris, Joe (1991). Representation theory. A first course. Graduate Texts in Mathematics, Readings in Mathematics. Vol. 129. New York: Springer-Verlag. doi:10.1007/978-1-4612-0979-9. ISBN 978-0-387-97495-8. MR 1153249. OCLC 246650103.
• Jacobson, Nathan, Lie algebras, Republication of the 1962 original. Dover Publications, Inc., New York, 1979. ISBN 0-486-63832-4; Chapter X considers a classification of simple Lie algebras over a field of characteristic zero.

Category:Lie algebras