Lie algebras

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia

Last update: 9 September 2012

Lie algebras

A Lie algebra over a field F is a vector space 𝔤 over F with a bracket [,]: 𝔤×𝔤𝔤 which is bilinear and satisfies

  1. [x,y] =-[y,x], for all x,y𝔤,
  2. (Jacobi identity) [x,[y,z]] + [z,[x,y]] + [y,[z,x]] =0 , for all x,y,z𝔤.
The derived series of 𝔤 is the sequence D0𝔤 D1𝔤 , where D0𝔤=𝔤   and   Di+1𝔤 =[Di𝔤 ,Di𝔤] .

The lower central series of 𝔤 is the sequence C1𝔤 C2𝔤, where C0𝔤=𝔤   and   Ci+1𝔤 =[𝔤, Ci𝔤] .

Let 𝔤 be a Lie algebra.

  1. 𝔤 is abelian if [𝔤,𝔤]=0 .
  2. 𝔤 is nilpotent if Cn(𝔤) =0 for all sufficiently large n.
  3. 𝔤 is solvable if Dn(𝔤)=0 for all sufficiently large n.
  4. The radical rad(𝔤) is the largest solvable ideal of 𝔤.
  5. The nilradical nil(𝔤) is the largest nilpotent?????? ideal of 𝔤.
  6. 𝔤 is semisimple if rad(𝔤)=0.
  7. 𝔤 is reductive if nil(𝔤)=0.
  8. A Cartan subalgebra is a maximal abelian subalgebra of semisimple elements.

Then 0nil(𝔤) rad(𝔤)𝔤 where nil(𝔤) is nilpotent, rad(𝔤) is solvable, 𝔤/ rad(𝔤) is semisimple, rad(𝔤) /nil(𝔤) is abelian, nil(𝔤) is nilpotent.

Example [Bou, Chap I, Section 4, Prop 5] The following are equivalent:

  1. 𝔤 is reductive.
  2. The adjoint representation of 𝔤 is semisimple.
  3. [𝔤,𝔤] is a semisimple Lie algebra,
  4. 𝔤 is the direct sum of a semisimple Lie algebra and a commutative Lie algebra.
  5. 𝔤 has a finite dimensional representation such that the associated bilinear form is nondegenerate.
  6. 𝔤 has a faithful finite dimensional representation.
  7. rad(𝔤) is the center of 𝔤.

HW: Show that 𝔤 is reductive if all its representations are completely decomposable.

HW: Show that 𝔤 is reductive if 𝔤 =Z(𝔤) [𝔤,𝔤] with [𝔤,𝔤] semisimple.

The finite dimensional simple Lie algebras over are

  1. (Type An-1) 𝔰𝔩n () , for n2,
  2. (Type Bn) 𝔰𝔬2n+1 (), for n1,
  3. (Type Cn) 𝔰𝔭2n (), for n1,
  4. (Type Dn) 𝔰𝔬2n (), for ,n4, and
  5. the five simple Lie algebras E6, E7, E8, F4, G2.

The finite dimensional simple Lie algebras over are ???????????????????????

Notes and References

This page is a retyping of notes from ????


[Bou] N. Bourbaki, Lie groups and Lie algebras, Masson, Paris???? ISBN: ?????? MR??????.

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