Signed Tensor Product Functorial Construction¶
AUTHORS:
- Travis Scrimshaw (2019-07): initial version 
- class sage.categories.signed_tensor.SignedTensorProductFunctor[source]¶
- Bases: - CovariantFunctorialConstruction- 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- B, then- A.SignedTensorProducts()is a subcategory of- B.SignedTensorProducts()(see also- 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 - SignedTensorProductsof- Algebras(QQ).Graded(). This nested class is itself a subclass of- SignedTensorProductsCategory.- EXAMPLES: - sage: tensor_signed The signed tensor functorial construction - >>> from sage.all import * >>> tensor_signed The signed tensor functorial construction - symbol = ' # '¶
 - unicode_symbol = ' ⊗ '¶
 
- class sage.categories.signed_tensor.SignedTensorProductsCategory(category, *args)[source]¶
- Bases: - CovariantConstructionCategory- An abstract base class for all SignedTensorProducts’s categories. - SignedTensorProducts()[source]¶
- Return the category of signed tensor products of objects of - self.- 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)- EXAMPLES: - sage: AlgebrasWithBasis(QQ).Graded().SignedTensorProducts().SignedTensorProducts() Category of signed tensor products of graded algebras with basis over Rational Field - >>> from sage.all import * >>> AlgebrasWithBasis(QQ).Graded().SignedTensorProducts().SignedTensorProducts() Category of signed tensor products of graded algebras with basis over Rational Field 
 - base()[source]¶
- The base of a signed tensor product is the base (usually a ring) of the underlying category. - EXAMPLES: - sage: AlgebrasWithBasis(ZZ).Graded().SignedTensorProducts().base() Integer Ring - >>> from sage.all import * >>> AlgebrasWithBasis(ZZ).Graded().SignedTensorProducts().base() Integer Ring