The notion of a Lie algebra is first established in general. Let
be any vector space with coefficients in
. In , the vectors
can be added or multiplied by elements of
; however, there is no
way to ``multiply'' two vectors to obtain a third. The Lie algebra
introduces a product operation to . The product is called a
*bracket* or *Lie bracket* (considered here as a
generalization of the previous Lie bracket) and is denoted by
.

To be a *Lie algebra* obtained from , the bracket must satisfy
the following three axioms:

**Bilinearity:**For any and ,(15.95)

**Skew symmetry:**For any ,(15.96)

This means that the bracket is anti-commutative.**Jacobi identity:**For any ,(15.97)

Note that the bracket is not even associative.

(15.98) |

It can be verified that the required axioms of a Lie bracket are satisfied.

One interesting property of the cross product that is exploited often in analytic geometry is that it produces a vector outside of the span of and . For example, let be the two-dimensional subspace of vectors

(15.99) |

The cross product always yields a vector that is a multiple of , which lies outside of if the product is nonzero. This behavior is very similar to constructing vector fields that lie outside of using the Lie bracket in Section 15.4.2.

- The product , interpreted as a function from , is smooth.
- The inverse , interpreted as a function from to , is smooth.

For any Lie group, a Lie algebra can be defined on a special set of
vector fields. These are defined using the *left translation*
mapping
. The vector field formed from the
differential of is called a *left-invariant vector field*.
A Lie algebra can be defined on the set of these fields.
The Lie bracket definition depends on the particular group. For the
case of , the Lie bracket is

(15.100) |

In this case, the Lie bracket clearly appears to be a test for commutativity. If the matrices commute with respect to multiplication, then the Lie bracket is zero. The Lie brackets for and are given in many texts on mechanics and control [156,846]. The Lie algebra of left-invariant vector fields is an important structure in the study of nonlinear systems and mechanics.

Steven M LaValle 2012-04-20