Simple Lie algebra

Lie groups and Lie algebras
Classical groups General linear GL(n) Special linear SL(n) Orthogonal O(n) Special orthogonal SO(n) Unitary U(n) Special unitary SU(n) Symplectic Sp(n)
Simple Lie groups Classical An Bn Cn Dn Exceptional G2 F4 E6 E7 E8
Other Lie groups Circle Lorentz Poincaré Conformal group Diffeomorphism Loop Euclidean
Lie algebras Lie group–Lie algebra correspondence Exponential map Adjoint representation Killing form Index Simple Lie algebra Loop algebra Affine Lie algebra
Semisimple Lie algebra Dynkin diagrams Cartan subalgebra Root system Weyl group Real form Complexification Split Lie algebra Compact Lie algebra
Representation theory Lie group representation Lie algebra representation Representation theory of semisimple Lie algebras Representations of classical Lie groups Theorem of the highest weight Borel–Weil–Bott theorem
Lie groups in physics Particle physics and representation theory Lorentz group representations Poincaré group representations Galilean group representations
Scientists Sophus Lie Henri Poincaré Wilhelm Killing Élie Cartan Hermann Weyl Claude Chevalley Harish-Chandra Armand Borel
Glossary Table of Lie groups
v t e

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

A direct sum of simple Lie algebras is called a semisimple Lie algebra.

A simple Lie group is a connected Lie group whose Lie algebra is simple.

Complex simple Lie algebras

A finite-dimensional simple complex Lie algebra is isomorphic to either of the following: ${\displaystyle {\mathfrak {sl}}_{n}\mathbb {C} }$, ${\displaystyle {\mathfrak {so}}_{n}\mathbb {C} }$, ${\displaystyle {\mathfrak {sp}}_{2n}\mathbb {C} }$ (classical Lie algebras) or one of the five exceptional Lie algebras.^[1]

To each finite-dimensional complex semisimple Lie algebra ${\displaystyle {\mathfrak {g}}}$, there exists a corresponding diagram (called the Dynkin diagram) where the nodes denote the simple roots, the nodes are jointed (or not jointed) by a number of lines depending on the angles between the simple roots and the arrows are put to indicate whether the roots are longer or shorter.^[2] The Dynkin diagram of ${\displaystyle {\mathfrak {g}}}$ is connected if and only if ${\displaystyle {\mathfrak {g}}}$ is simple. All possible connected Dynkin diagrams are the following:^[3]

where n is the number of the nodes (the simple roots). The correspondence of the diagrams and complex simple Lie algebras is as follows:^[2]

(A_n) ${\displaystyle \quad {\mathfrak {sl}}_{n+1}\mathbb {C} }$

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

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

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

The rest, exceptional Lie algebras.

Real simple Lie algebras

If ${\displaystyle {\mathfrak {g}}_{0}}$ is a finite-dimensional real simple Lie algebra, its complexification is either (1) simple or (2) a product of a simple complex Lie algebra and its conjugate. For example, the complexification of ${\displaystyle {\mathfrak {sl}}_{n}\mathbb {C} }$ thought of as a real Lie algebra is ${\displaystyle {\mathfrak {sl}}_{n}\mathbb {C} \times {\overline {{\mathfrak {sl}}_{n}\mathbb {C} }}}$. Thus, a real simple Lie algebra can be classified by the classification of complex simple Lie algebras and some additional information. This can be done by Satake diagrams that generalize Dynkin diagrams. See also Table of Lie groups#Real Lie algebras for a partial list of real simple Lie algebras.

Notes

See also

• Simple Lie group
• Vogel plane

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.

• "Lie algebra, semi-simple", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
• Simple Lie algebra at the nLab

This page was last edited on 11 October 2023, at 09:30