Classical group

In mathematics, the classical groups are defined as the special linear groups over the reals ${\displaystyle \mathbb {R} }$, the complex numbers ${\displaystyle \mathbb {C} }$ and the quaternions ${\displaystyle \mathbb {H} }$ together with special^[1] automorphism 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.^[2] 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.^[3]