Other matrix groups

Here we summarize some other common matrix groups. We again stress that although they are defined in terms of matrices with non-real elements, these are all real Lie groups.

Name Matrix Lie algebra
\({GL\left(n,\mathbb{C}\right)}\) \({n\times n}\) complex matrices with \({\textrm{det}\left(A\right)\neq0}\) All complex \({n\times n}\) matrices
\({SL\left(n,\mathbb{C}\right)}\) \({\textrm{det}\left(A\right)=1}\) \({\textrm{tr}\left(A\right)=0}\)
\({U\left(n\right)}\)
Unitary group
\({A^{\dagger}A=AA^{\dagger}=I}\) \({\Rightarrow\left|\textrm{det}\left(A\right)\right|=1}\) \({A^{\dagger}=-A}\)
\({SU\left(n\right)}\)
Special Unitary group
\({A^{\dagger}A=AA^{\dagger}=I,}\) \({\textrm{det}\left(A\right)=1}\) \({A^{\dagger}=-A,}\)
\({\textrm{tr}\left(A\right)=0}\)
\({Sp\left(2n,\mathbb{C}\right)}\)
Complex symplectic group
\({A^{\textrm{T}}JA=J}\) \({JA+A^{\textrm{T}}J=0}\)
\({Sp\left(n\right)}\)
Quaternionic symplectic group
\({n\times n}\) quaternionic matrices with \({A^{\dagger}A=AA^{\dagger}=I}\) where \({^{\dagger}}\) uses the quaternionic conjugate \({A^{\dagger}=-A}\)

Notes: \({U(n)}\) is the complex version of \({O(n)}\), and can be viewed as preserving the standard inner product \({\left\langle v,w\right\rangle \equiv v^{\dagger}w}\) on \({\mathbb{C}^{n}}\); however it does not form a complex Lie group. The complex versions of the pseudo-orthogonal and special pseudo-orthogonal groups can be similarly defined.

The quaternionic symplectic group \({Sp(n)}\) is also called the quaternionic unitary group, which better matches the definition above. An equivalent definition is \({Sp\left(n\right)\equiv U\left(2n\right)\cap Sp\left(2n,\mathbb{C}\right)}\), and thus \({Sp(n)}\) is also called the unitary symplectic group. Unlike in the real and complex cases, it is also compact, and so yet another term used is the compact symplectic group. One can also view the relationships between the three symplectic groups in terms of their Lie algebras; this will be seen in the section on simple Lie algebras.

Additional matrix groups can be defined by generalizing more of the above constructions to mixed signatures and quaternionic entries, but they are not as frequently used in physics and we will not cover them here.

An Illustrated Handbook