Signed Tensor Product Functorial Construction¶
Travis Scrimshaw (2019-07): initial version
- class sage.categories.signed_tensor.SignedTensorProductFunctor¶
A singleton class for the signed tensor functor.
This functor takes a collection of graded algebras (possibly with basis) and constructs the signed tensor product of those algebras. If this algebra is in a subcategory, say that of
Algebras(QQ).Graded(), it is automatically endowed with its natural algebra structure, thanks to the category
Algebras(QQ).Graded().SignedTensorProducts()of signed tensor products of graded algebras. For elements, it constructs the natural tensor product element in the corresponding tensor product of their parents.
The signed tensor functor is covariant: if
Ais a subcategory of
A.SignedTensorProducts()is a subcategory of
CovariantFunctorialConstruction). Hence, the role of
Algebras(QQ).Graded().SignedTensorProducts()is solely to provide mathematical information and algorithms which are relevant to signed tensor product of graded algebras.
Those are implemented in the nested class
Algebras(QQ).Graded(). This nested class is itself a subclass of
sage: tensor_signed The signed tensor functorial construction
- class sage.categories.signed_tensor.SignedTensorProductsCategory(category, *args)¶
An abstract base class for all SignedTensorProducts’s categories.
Return the category of signed tensor products of objects of
By associativity of signed tensor products, this is
self(a tensor product of signed tensor products of \(Cat\)’s is a tensor product of \(Cat\)’s with the same twisting morphism)
sage: AlgebrasWithBasis(QQ).Graded().SignedTensorProducts().SignedTensorProducts() Category of signed tensor products of graded algebras with basis over Rational Field
The base of a signed tensor product is the base (usually a ring) of the underlying category.
sage: AlgebrasWithBasis(ZZ).Graded().SignedTensorProducts().base() Integer Ring
- sage.categories.signed_tensor.tensor_signed = The signed tensor functorial construction¶