Symplectic group

In mathematics, the term symplectic group can refer to two different, but closely related types of groups , denoted by Sp (2 n , F ) and Sp ( n ). The latter are sometimes called compact symplectic groups , unlike the former. Slightly different notations are used, especially significant differences relate to the presence or absence of the factor 2 in the notation.

The symplectic group Sp (2 n , F ) consists of 2 n × 2 n symplectic matrices equipped with the usual matrix multiplication, with elements from the field F.

The term was introduced by Hermann Weil to replace the old confusing names, and is a Greek word meaning complex (replacing complex that was used earlier in Europe, which leads to confusion in European languages).