Lie algebra

A Lie algebra is a Vector space equipped with a product ⋅\cdot with the following properties:

  1. It is bilinear: a⋅(βb⋅γc)=βa⋅b+γa⋅ca\cdot(\beta b\cdot \gamma c)=\beta a\cdot b+\gamma a\cdot c.
  2. It is antisymmetric: a⋅b=−b⋅aa\cdot b=-b\cdot a.
  3. It satisfies the Jacobi identity: a⋅(b⋅c)+b⋅(c⋅a)+c⋅(a⋅b)=0a\cdot(b\cdot c)+b\cdot(c\cdot a)+c\cdot(a\cdot b)=0.

Generally speaking, this operation is either the Poisson brackets or the Commutator.

Examples

The space of all rotations is a Lie algebra.