simple Lie algebra
-Lie theory ∞
∞-Lie theory Background
Formal Lie groupoids
-Lie groupoids ∞
-Lie groups ∞
-Lie algebroids ∞
-Lie algebras ∞ Simple Lie algebras
simple Lie algebra is a Lie algebra such that: 𝔤
Equivalently, a simple Lie algebra is a
simple object of LieAlg that also is nonabelian. Note that there are only two abelian Lie algebras whose only proper ideal is the zero ideal: the trivial Lie algebra (which is not a simple object in either, since the zero ideal is not proper either) and the Lie Alg line (which is a simple object in but is still not considered a simple Lie algebra). Lie Alg Classification
Simple Lie algebras over an
algebraically closed field of characteristic zero, like many other things in mathematics, may be classified by Dynkin diagrams. We have: ?
, the 𝔞 n = 𝔰𝔩 n + 1 special linear Lie algebra of rank ? . We count this only for n , since n ≥ 1 is the 𝔞 0 trivial Lie algebra (which is not simple but is still semisimple).
, the odd-dimensional 𝔟 n = 𝔰𝔬 2 n + 1 special orthogonal Lie algebra of rank . We count this only for n , since n ≥ 2 for 𝔟 n = 𝔞 n . n < 2
, the 𝔠 n = 𝔰𝔭 n symplectic Lie algebra of rank ? . We count this only for n , since n ≥ 3 for 𝔠 n = 𝔟 n . n < 3
, the even-dimensional 𝔡 n = 𝔰𝔬 2 n special orthogonal Lie algebra of rank . We count this only for n , since n ≥ 4 for 𝔡 n = 𝔞 n and n < 2 , while n = 3 (which is not simple but is still 𝔡 2 = 𝔞 2 ⊕ 𝔞 2 semisimple).
, an 𝔢 n exceptional Lie algebra that only exists for rank ? . We count this only for n < 9 (thus for n ≥ 6 in all), since n = 6 , 7 , 8 , 𝔢 5 = 𝔡 5 , 𝔢 4 = 𝔞 4 (which is not simple but is still 𝔢 n = 𝔞 n − 1 ⊕ 𝔞 1 semisimple) for , and 2 ≤ n < 4 for 𝔢 n = 𝔞 n . n < 2
exceptional Lie algebras ? and 𝔣 4 , which exist only for those ranks. 𝔤 2
If you want to classify simple objects in
, then there is one other possibility: the Lie Alg line (which has no corresponding Dynkin diagram).
It is much more difficult to classify simple Lie algebras over non-closed fields, over fields with positive characteristic, and especially over non-fields.
Semisimple Lie algebras
is a semisimple Lie algebra direct sum of simple Lie algebras. In particular, every simple Lie algebra is semisimple, but there are many more. Simple Lie groups
Lie group is a simple Lie group if the Lie algebra corresponding to it under Lie integration is simple.
Revised on September 7, 2010 05:22:13