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simple Lie algebra

Context

\infty-Lie theory

∞-Lie theory

Background

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Lie theory

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\infty-Lie groupoids

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\infty-Lie algebras

Simple Lie algebras

Definition

A simple Lie algebra is a Lie algebra 𝔤\mathfrak{g} such that:

  • 𝔤\mathfrak{g} is not abelian, and
  • the only proper ideal of 𝔤\mathfrak{g} is the zero ideal.

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 LieAlgLie Alg either, since the zero ideal is not proper either) and the line (which is a simple object in LieAlgLie Alg but is still not considered a simple Lie algebra).

Classification

Simple Lie algebras over an algebraically closed field of characteristic zero, like many other things in mathematics, may be classified by Dynkin diagram?s. We have:

  • 𝔞 n=𝔰𝔩 n+1\mathfrak{a}_n = \mathfrak{sl}_{n+1}, the special linear Lie algebra? of rank nn. We count this only for n1n \geq 1, since 𝔞 0\mathfrak{a}_0 is the trivial Lie algebra (which is not simple but is still semisimple).

  • 𝔟 n=𝔰𝔬 2n+1\mathfrak{b}_n = \mathfrak{so}_{2n+1}, the odd-dimensional special orthogonal Lie algebra of rank nn. We count this only for n2n \geq 2, since 𝔟 n=𝔞 n\mathfrak{b}_n = \mathfrak{a}_n for n<2n \lt 2.

  • 𝔠 n=𝔰𝔭 n\mathfrak{c}_n = \mathfrak{sp}_n, the symplectic Lie algebra? of rank nn. We count this only for n3n \geq 3, since 𝔠 n=𝔟 n\mathfrak{c}_n = \mathfrak{b}_n for n<3n \lt 3.

  • 𝔡 n=𝔰𝔬 2n\mathfrak{d}_n = \mathfrak{so}_{2n}, the even-dimensional special orthogonal Lie algebra of rank nn. We count this only for n4n \geq 4, since 𝔡 n=𝔞 n\mathfrak{d}_n = \mathfrak{a}_n for n<2n \lt 2 and n=3n = 3, while 𝔡 2=𝔞 2𝔞 2\mathfrak{d}_2 = \mathfrak{a}_2 \oplus \mathfrak{a}_2 (which is not simple but is still semisimple).

  • 𝔢 n\mathfrak{e}_n, an exceptional Lie algebra? that only exists for rank n<9n \lt 9. We count this only for n6n \geq 6 (thus for n=6,7,8n = 6, 7, 8 in all), since 𝔢 5=𝔡 5\mathfrak{e}_5 = \mathfrak{d}_5, 𝔢 4=𝔞 4\mathfrak{e}_4 = \mathfrak{a}_4, 𝔢 n=𝔞 n1𝔞 1\mathfrak{e}_n = \mathfrak{a}_{n-1} \oplus \mathfrak{a}_1 (which is not simple but is still semisimple) for 2n<42 \leq n \lt 4, and 𝔢 n=𝔞 n\mathfrak{e}_n = \mathfrak{a}_n for n<2n \lt 2.

  • the exceptional Lie algebra?s 𝔣 4\mathfrak{f}_4 and 𝔤 2\mathfrak{g}_2, which exist only for those ranks.

If you want to classify simple objects in LieAlgLie Alg, then there is one other possibility: the 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

A semisimple Lie algebra is a direct sum of simple Lie algebras. In particular, every simple Lie algebra is semisimple, but there are many more.

Simple Lie groups

A 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 by Urs Schreiber (134.100.32.213)