Lie algebra

Definition

A Lie algebra over a field $k$ is defined as a set $V$ equipped with the following structures:

• A vector space structure over $k$
• A $k$-bilinear map $[,]:V\times V\to V$ called the Lie bracket

satisfying the following compatibility conditions:

• $[x,x]=0\mathrm{\forall }x\in V$
• $[x,y]+[y,x]=0\mathrm{\forall }x,y\in V$
• $[[x,y],z]+[[y,z],x]+[[z,x],y]=0\mathrm{\forall }x,y,z\in V$ (this is called the Jacobi identity)

We note that the first condition will imply the second ( take $x+y$ for $x$) but the second will imply the first only when the field is of characteristic 2.

Facts

Lie algebra of a Lie group

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Universal enveloping algebra

Further information: Universal enveloping algebra

Every Lie algebra has a universal enveloping algebra. An enveloping algebra for a Lie algebra is an associative algebra over the same base field which contains the elements of the Lie algebra, such that:

• The addition in the enveloping algebra is the same as that within the Lie algebra
• For those elements which are in the Lie algebra, the commutator coincides with the Lie bracket

The universal enveloping algebra is an algebra that is universal among all enveloping algebras.