Lie Algebra

A Lie algebra is a vector space over a field with a bilinear Lie bracket that is alternating and satisfies the Jacobi identity. Elements , , and of a Lie algebra satisfy

(1)

(2)

and

(3)

(the Jacobi identity). The relation implies

(4)

For characteristic not equal to two, these two relations are equivalent.

For the Poisson bracket of functions, the additional Leibniz rule is

(5)

An associative algebra with associative product can be made into a Lie algebra by the Lie product

(6)

Every Lie algebra is isomorphic to a subalgebra of some where the associative algebra may be taken to be the linear operators over a vector space (the Poincaré-Birkhoff-Witt theorem; Jacobson 1979, pp. 159-160). If is finite dimensional, then can be taken to be finite dimensional (Ado's theorem for characteristic ; Iwasawa's theorem for characteristic ).

The classification of finite dimensional simple Lie algebras over an algebraically closed field of characteristic 0 can be accomplished by (1) determining matrices called Cartan matrices corresponding to indecomposable simple systems of roots and (2) determining the simple algebras associated with these matrices (Jacobson 1979, p. 128). This is one of the major results in Lie algebra theory, and is frequently accomplished with the aid of diagrams called Dynkin diagrams.

A complex Lie algebra need not have a real form, even when it is isomorphic to its complex conjugate. De Medts (2026) constructed such a nine-dimensional nilpotent Lie algebra of nilpotency class five, generated by two elements, with Claude assistance in the construction.