Jacobi group
Encyclopedia
In mathematics, the Jacobi group, introduced by
, is the semidirect product
of the symplectic group
Sp2n(R) and the Heisenberg group R1+2n. The concept is name after Carl Gustav Jacob Jacobi. Automorphic form
s on the Jacobi group are called Jacobi forms.
, is the semidirect product
Semidirect product
In mathematics, specifically in the area of abstract algebra known as group theory, a semidirect product is a particular way in which a group can be put together from two subgroups, one of which is a normal subgroup. A semidirect product is a generalization of a direct product...
of the symplectic group
Symplectic group
In mathematics, the name symplectic group can refer to two different, but closely related, types of mathematical groups, denoted Sp and Sp. The latter is sometimes called the compact symplectic group to distinguish it from the former. Many authors prefer slightly different notations, usually...
Sp2n(R) and the Heisenberg group R1+2n. The concept is name after Carl Gustav Jacob Jacobi. Automorphic form
Automorphic form
In mathematics, the general notion of automorphic form is the extension to analytic functions, perhaps of several complex variables, of the theory of modular forms...
s on the Jacobi group are called Jacobi forms.