The most common Lie groups are the matrix groups (AKA linear groups), which are Lie subgroups of the group of real or complex \({n\times n}\) invertible matrices, denoted \({GL(n,\mathbb{R})}\) and \({GL(n,\mathbb{C})}\). We can also consider the linear groups, which are Lie subgroups of \({GL(V)}\), the group of invertible linear transformations on a real or complex vector space \({V}\). One can then choose a basis of \({V}\) to get a (non-canonical) isomorphism to the matrix group, e.g. from \({GL(\mathbb{R}^{n})}\) to \({GL(n,\mathbb{R})}\).
| Δ The distinction between the abstract linear groups and the basis-dependent matrix groups is not always made, and the notation is used interchangeably. Alternative notation includes \({GL_{n}(\mathbb{R})}\), and the field and/or dimension is often omitted, yielding notation such as \({GL_{n}}\), \({GL(n)}\), \({GL(\mathbb{R})}\), or \({GL}\). |
These groups can be seen to be Lie groups by taking global coordinates to be the real matrix entries or the real components of the complex entries. Thus \({GL(n,\mathbb{R})}\) is a manifold of dimension \({n^{2}}\), and \({GL(n,\mathbb{C})}\) has manifold dimension \({2n^{2}}\). Any subgroup of \({GL}\) that is also a submanifold is then automatically a Lie subgroup.
We can also consider Lie groups defined by invertible matrices with entries in \({\mathbb{H}}\) or \({\mathbb{O}}\), since even though they cannot be viewed as linear transformations on a vector space, they still form a group and are manifolds with respect to the real components of their entries.
| Δ Some matrix groups can also be viewed as a complex Lie group, a group that is also a complex manifold. For example, \({GL(n,\mathbb{C})}\) can be viewed as an \({n^{2}}\)-dimensional complex Lie group instead of as a real Lie group of dimension \({2n^{2}}\). It is important to distinguish between a complex Lie group and a real Lie group defined by matrices with complex entries. |