Lie Algebra
A Lie algebra over a field
is a -vector space endowed with an operation which is linear in each argument, satisfies for all and the Jacobi identity: for all .
e.g.
- If
is a smooth manifold, is a Lie algebra. - If
is a finite dimensional vector space over a field of dimension , then the space of endomorphisms (i.e. the set of all linear maps ) is a Lie algebra over with the Lie bracket defined as the commutator: Only the Jacobi identity needs to be checked: To distinguish this Lie algebra from the ring , we will denote it as or , and call it the general linear Lie algebra of .
Proposition
Lie algebra is a non-associative algebra over the field.
Proof Lie bracket gives a multiplication which is compatible with the vector space structure. i.e. Lie bracket is bilinear.
The concepts of isomorphism and Lie subalgebra are defined in a familiar way:
Isomorphic Lie Algebras
Two Lie algebras
and are said to be isomorphic if there exists a linear bijection such that for all .
Lie Subalgebra
A Lie subalgebra of a Lie algebra
is a subspace such that for all .