Classical Group - Relationship With Bilinear Forms

Relationship With Bilinear Forms

The unifying feature of classical Lie groups is that they are close to the isometry groups of certain bilinear or sesquilinear forms. The four series are labelled by the Dynkin diagram attached to them, with subscript n ≥ 1. The families may be represented as follows:

• A_n = SU(n + 1), the special unitary group of unitary n+1-by-n+1 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.

Viewing a classical group G as a subgroup of GL(n) via its definition as automorphisms of a vector space preserving some involution provides a representation of G called the standard representation.