Lie Algebras — Studies in Algebra and Group Theory

17.13 Lie Algebras

The idea: a Lie group is a curved object (a manifold), but its tangent space at the identity is a flat object (a vector space) that retains much of the group’s structure. This tangent space is the Lie algebra.

Definition 17.13.1 (Lie algebra (abstract)). A Lie algebra over a field is a -vector space equipped with a bilinear operation (the Lie bracket) satisfying:

Antisymmetry: for all (equivalently, ).
The Jacobi identity: for all .

Definition 17.13.2 (Lie algebra of a matrix Lie group). For a matrix Lie group , the Lie algebra is the tangent space to at the identity. This is a real vector space with the commutator bracket .

Note. The commutator bracket satisfies the Lie algebra axioms: antisymmetry is clear (), and the Jacobi identity follows by expanding the triple brackets.

Example 17.13.1 (Classical Lie algebras).

(all real matrices), dimension .
, dimension . (Differentiating gives .)
(skew-symmetric matrices), dimension . (Differentiating gives .)
(skew-Hermitian traceless), dimension .
, dimension .