Matrix Lie Groups — Studies in Algebra and Group Theory

17.12 Matrix Lie Groups

The finite groups studied so far (symmetric groups, dihedral groups, etc.) are discrete. Many important groups in mathematics and physics are continuous: the group of rotations , the group of invertible matrices , and so on. Lie theory provides the algebraic tools to study these.

Definition 17.12.1 (Matrix Lie group). A matrix Lie group is a closed subgroup (closed in the topology inherited from ). The “Lie” condition means is simultaneously a group and a smooth manifold, with the group operations (multiplication and inversion) being smooth maps.

Example 17.12.1 (The classical matrix Lie groups).

, : the general linear groups (invertible matrices).
, : the special linear groups.
: the orthogonal group. : the special orthogonal group (rotations).
: the unitary group. .
where : the symplectic group.

Note. The dimension of a Lie group (as a manifold) can be computed from its defining equations: has dimension ; the constraint removes 1 dimension, giving ; the constraint gives equations (since is symmetric), so .