Lie algebra

The concept of a Lie algebra is closely tied to a Lie group, but they are separate. Every Lie group has an associated Lie algebra, but the definition of a Lie algebra can be taken and used independently.

Definition

A Lie algebra is a vector space with a map called the Lie bracket. The Lie bracket is a map $[\cdot ,\cdot ]:\mathfrak{g}\times \mathfrak{g}\to \mathfrak{g}$ that satisfies the following properties $\forall X,Y,Z\in \mathfrak{g}$ :

• Bilinearity: $[aX+bY,Z]=a[X,Z]+b[Y,Z]$ (and also in the second component.)
• Antisymmetry: $[X,Y]=-[Y,X]$
• Jacobi Indentity: $[X,[Y,Z]]+[Y,[Z,X]]+[Z,[X,Y]]=0$

Note that a Lie algebra is NOT(!!) an associative algebra, the Jacobi identity is a type of substitute for associativity.

Morphisms

A Lie algebra homomorphism is a linear map $\varphi :\mathfrak{g}\to \mathfrak{h}$ that preserves the Lie bracket, i.e. $\varphi ([X,Y])=[\varphi (X),\varphi (Y)]\forall X,Y\in \mathfrak{g}$

Subalgebra

A Lie subalgebra is linear space $\mathfrak{h}\subseteq \mathfrak{g}$ that is closed under the Lie bracket.

Important constructions