Classical group

In mathematics, the classical groups are the matrix groups arising from finite-dimensional vector spaces and from nondegenerate bilinear, sesquilinear, quadratic, and Hermitian forms. In the traditional setting of Lie groups, this includes the real, complex, and quaternionic general linear, special linear, orthogonal, unitary, and symplectic groups, together with their indefinite analogues.

In the language of linear algebraic groups, the connected classical groups are the connected reductive groups of Dynkin types

A

n

{\displaystyle A_{n}}

,

B

n

{\displaystyle B_{n}}

,

C

n

{\displaystyle C_{n}}

, and

D

n

{\displaystyle D_{n}}

, together with their forms over arbitrary fields. Over

R

{\displaystyle \mathbb {R} }

and

C

{\displaystyle \mathbb {C} }

this recovers the familiar classical Lie groups, while over finite fields one obtains the finite classical groups.

The term goes back to Hermann Weyl's book The Classical Groups. Among the simple Lie groups, the classical groups are in contrast to the exceptional Lie groups, G2, F4, E6, E7, E8, which share their abstract properties, but not their familiarity.

This article begins with the classical Lie groups over

R

{\displaystyle \mathbb {R} }

,

C

{\displaystyle \mathbb {C} }

, and

H

{\displaystyle \mathbb {H} }

, and later discusses the more general formulation over arbitrary fields.

Overview

Two closely related usages of the term classical group occur in the literature. In the older matrix-group literature, classical groups are the linear groups over

R

{\displaystyle \mathbb {R} }

,

C

{\displaystyle \mathbb {C} }

, and

H

{\displaystyle \mathbb {H} }

together with the groups preserving nondegenerate forms on those spaces. In the modern theory of algebraic groups, the phrase usually refers to the groups of types

A

{\displaystyle A}

,

B

{\displaystyle B}

,

C

{\displaystyle C}

, and

D

{\displaystyle D}

and their forms over general fields.