Classical groups

In mathematics, the classical groups are defined as the special linear groups over the reals $R$ , the complex numbers $C$ and the quaternions $H$ together with specialautomorphism groups of symmetric or skew-symmetric bilinear forms and Hermitian or skew-Hermitian sesquilinear forms defined on real, complex and quaternionic finite-dimensional vector spaces. Of these, the complex classical Lie groups are four infinite families of Lie groups that together with the exceptional groups exhaust the classification of simple Lie groups. The compact classical groups are compact real forms of the complex classical groups. The finite analogues of the classical groups are the classical groups of Lie type. The term "classical group" was coined by Hermann Weyl, it being the title of his 1939 monograph The Classical Groups.

The classical groups form the deepest and most useful part of the subject of linear Lie groups. Most types of classical groups find application in classical and modern physics. A few examples are the following. The rotation group $SO(3)$ is a symmetry of Euclidean space and all fundamental laws of physics, the Lorentz group $O(3,1)$ is a symmetry group of spacetime of special relativity. The special unitary group $SU(3)$ is the symmetry group of quantum chromodynamics and the symplectic group $Sp(m)$ finds application in hamiltonian mechanics and quantum mechanical versions of it.