Lie Algebra - Definition and First Properties

Definition and First Properties

A Lie algebra is a vector space over some field F together with a binary operation called the Lie bracket, which satisfies the following axioms:

• Bilinearity:

for all scalars a, b in F and all elements x, y, z in .

• Alternating on :

for all x in .

• The Jacobi identity:

for all x, y, z in .

Note that the bilinearity and alternating properties imply anticommutativity, i.e., for all elements x, y in, while anticommutativity only implies the alternating property if the field's characteristic is not 2.

For any associative algebra A with multiplication, one can construct a Lie algebra L(A). As a vector space, L(A) is the same as A. The Lie bracket of two elements of L(A) is defined to be their commutator in A:

The associativity of the multiplication * in A implies the Jacobi identity of the commutator in L(A). In particular, the associative algebra of n × n matrices over a field F gives rise to the general linear Lie algebra The associative algebra A is called an enveloping algebra of the Lie algebra L(A). It is known that every Lie algebra can be embedded into one that arises from an associative algebra in this fashion. See universal enveloping algebra.