Classical Lie group

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The classical Lie groups are four infinite families of Lie groups closely related to the symmetries of Euclidean spaces. There is a certain leeway in using the term classical group depending on the context, but the uniting feature of classical Lie groups is that they are close to the isometry groups of a certain bilinear or sesquilinear forms.

A_n = SU(n), the special unitary group of unitary n-by-n complex matrices with determinant 1.
B_n = SO(2n+1), the special orthogonal group of orthogonal 2n+1-by-2n+1 real matrices with determinant 1.
C_n = Sp(n), the symplectic group of n-by-n quaternionic matrices that preserve the usual inner product on Hⁿ.
D_n = SO(2n), the special orthogonal group of orthogonal 2n-by-2n real matrices with determinant 1.

For certain purposes it is also natural to drop the condition that the determinant be 1 and consider unitary groups and (disconnected) orthogonal groups. The table lists the so-called connected compact real forms of the groups; they have closely-related complex analogues and various non-compact forms, for example, together with compact orthogonal groups one considers indefinite orthogonal groups. The Lie algebras corresponding to these groups are known as the classical Lie algebras.

The term classical group seems to have been coined by Hermann Weyl (as seen in the title of his 1940 monograph). It probably reflects their relation to "classical" geometry, in the spirit of Felix Klein's Erlangen program.

Contrasting with the classical Lie groups are the exceptional Lie groups, which share their abstract properties, but not their familiarity.