Signed Tensor Product Functorial Construction

AUTHORS:

• Travis Scrimshaw (2019-07): initial version

class sage.categories.signed_tensor.SignedTensorProductFunctor

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 A is 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 SignedTensorProducts of Algebras(QQ).Graded(). This nested class is itself a subclass of SignedTensorProductsCategory.

EXAMPLES:

sage: tensor_signed
The signed tensor functorial construction

symbol = ' # '

unicode_symbol = ' ⊗ '

class sage.categories.signed_tensor.SignedTensorProductsCategory(category, *args)

Bases: CovariantConstructionCategory

An abstract base class for all SignedTensorProducts’s categories.

SignedTensorProducts()

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

base()

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