symplectic group

English

Etymology

So named by German mathematician Hermann Weyl, replacing previous confusing names. More at symplectic.

Noun

symplectic group (plural symplectic groups)

1. (linear algebra, group theory) For given field F and positive integer n, the group of 2n×2n symplectic matrices with elements in F.
   • 1985, Roger Howe, “Dual Pairs in Physics: Harmonic Oscillators, Photons, Electrons, and Singletons”, in Mosh Flato, Paul Sally, Gregg Zuckerman, editors, Applications of Group Theory in Physics and Mathematical Physics, American Mathematical Society, page 179:

     This representation ω arises by virtue of the existence of an action by automorphisms of the symplectic group on a certain two-step nilpotent group, commonly called the Heisenberg group.
     Weil was of course working in the adelic formalism of modern number theory, so he considered symplectic groups and Heisenberg groups with coefficients in a general local field.

   • 2001, G. Wassermann (translator), V. I. Arnol'd, A. B. Givental', Symplectic Geometry, V. I. Arnol'd, S. P. Novikov (editors), Dynamical Systems IV: Symplectic Geometry and its Applications, Springer, 2nd Edition, page 18,

     The exponential of an operator gives the exponential mapping $\textstyle H\mapsto {\text{exp}}(H)=\sum {H^{k}/k!}$ of the space of Hamiltonian operators to the symplectic group. The symplectic group acts by conjugation on itself and on its Lie algebra.

   • 2012, Rolf Berndt, Ralf Schmidt, Elements of the Representation Theory of the Jacobi Group, Springer, page v:

     The Jacobi group is a semidirect product of a symplectic group with a Heisenberg group.

Usage notes

Can be denoted Sp(2n, F), although other notations are also used. In particular, what is denoted Sp(2n, F) in some texts appears as Sp(n, F) in others.

Derived terms

• compact symplectic group

Further reading

• symplectic group on Wikipedia.Wikipedia
• Group of symplectic type on Wikipedia.Wikipedia