Hecke algebras¶
In Sage a “Hecke algebra” always refers to an algebra of endomorphisms of some explicit module, rather than the abstract Hecke algebra of double cosets attached to a subgroup of the modular group.
We distinguish between “anemic Hecke algebras”, which are algebras of Hecke operators whose indices do not divide some integer N (the level), and “full Hecke algebras”, which include Hecke operators coprime to the level. Morphisms in the category of Hecke modules are not required to commute with the action of the full Hecke algebra, only with the anemic algebra.
-
sage.modular.hecke.algebra.
AnemicHeckeAlgebra
¶ An anemic Hecke algebra, generated by Hecke operators with index coprime to the level.
-
sage.modular.hecke.algebra.
HeckeAlgebra
¶ A full Hecke algebra (including the operators \(T_n\) where \(n\) is not assumed to be coprime to the level).
-
sage.modular.hecke.algebra.
HeckeAlgebra_anemic
¶ An anemic Hecke algebra, generated by Hecke operators with index coprime to the level.
-
sage.modular.hecke.algebra.
HeckeAlgebra_base
¶ Base class for algebras of Hecke operators on a fixed Hecke module.
INPUT:
M
- a Hecke module
EXAMPLES:
sage: CuspForms(1, 12).hecke_algebra() # indirect doctest Full Hecke algebra acting on Cuspidal subspace of dimension 1 of Modular Forms space of dimension 2 for Modular Group SL(2,Z) of weight 12 over Rational Field
-
sage.modular.hecke.algebra.
HeckeAlgebra_full
¶ A full Hecke algebra (including the operators \(T_n\) where \(n\) is not assumed to be coprime to the level).
-
sage.modular.hecke.algebra.
is_HeckeAlgebra
(x)¶ Return True if x is of type HeckeAlgebra.
EXAMPLES:
sage: from sage.modular.hecke.algebra import is_HeckeAlgebra sage: is_HeckeAlgebra(CuspForms(1, 12).anemic_hecke_algebra()) True sage: is_HeckeAlgebra(ZZ) False