symplectic group

Noun

symplectic group (plural symplectic groups)

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