Filtered Algebras¶
- class sage.categories.filtered_algebras.FilteredAlgebras(base_category)¶
Bases:
sage.categories.filtered_modules.FilteredModulesCategory
The category of filtered algebras.
An algebra \(A\) over a commutative ring \(R\) is filtered if \(A\) is endowed with a structure of a filtered \(R\)-module (whose underlying \(R\)-module structure is identical with that of the \(R\)-algebra \(A\)) such that the indexing set \(I\) (typically \(I = \NN\)) is also an additive abelian monoid, the unity \(1\) of \(A\) belongs to \(F_0\), and we have \(F_i \cdot F_j \subseteq F_{i+j}\) for all \(i, j \in I\).
EXAMPLES:
sage: Algebras(ZZ).Filtered() Category of filtered algebras over Integer Ring sage: Algebras(ZZ).Filtered().super_categories() [Category of algebras over Integer Ring, Category of filtered modules over Integer Ring]
REFERENCES:
- class ParentMethods¶
Bases:
object
- graded_algebra()¶
Return the associated graded algebra to
self
.Todo
Implement a version of the associated graded algebra which does not require
self
to have a distinguished basis.EXAMPLES:
sage: A = AlgebrasWithBasis(ZZ).Filtered().example() sage: A.graded_algebra() Graded Algebra of An example of a filtered algebra with basis: the universal enveloping algebra of Lie algebra of RR^3 with cross product over Integer Ring