symplectic group

plural symplectic groups

1. group theoryFor given field F and positive integer n, the group of 2n×2n symplectic matrices with elements in F.

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

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

   2 more examples

   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.

   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

   The exponential of an operator gives the exponential mapping #92;textstyleH#92;mapsto#92;text#123;exp#125;(H)#61;#92;sum#123;Hᵏ#47;k#33;#125; of the space of Hamiltonian operators to the symplectic group. The symplectic group acts by conjugation on itself and on its Lie algebra.

   2001, V. I. Arnol'd, A. B. Givental', “Symplectic Geometry”, in G. Wassermann, transl., edited by V. I. Arnol'd and S. P. Novikov, Dynamical Systems IV: Symplectic Geometry and its Applications, 2nd edition, Springer, page 18