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