1.2 Lie algebras and other general properties

Every Lie group has an associated Lie algebra, which is the tangent space around the identity element of the group. That is, the Lie algebra is a vector space generated by differentiating the group transformations along chosen directions in the space, at the identity transformation. The basis elements of the Lie algebra (and thus of the tangent space) are called generators in this document. All tangent vectors represent linear combinations of the generators.

Importantly, the tangent space associated with a Lie group provides an ``optimal'' space in which to represent differential quantities related to the group. For instance, velocities, Jacobians, and covariances of transformations are well-represented in the tangent space around a transformation. This is the ``optimal'' space in which to represent differential quantities because

• The tangent space is a vector space with the same dimension as the number of degrees of freedom of the group transformations
• The exponential map converts any element of the tangent space exactly into a transformation in the group
• The adjoint representation linearly transforms tangent vectors from one tangent space to another

The adjoint property is what ensures that the tangent space has the same structure at all points on the manifold, because a tangent vector can always be transormed back to the tangent space around the identity.

Each Lie group described below also has a group action on 3D space. For instance, 3D rigid transformations have the action of rotating and translating points. The matrix representations given below make these actions explicit.