Meaning of symplectic group | Babel Free

Definitions

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

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.”

“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.”

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

CEFR level