of T[i][1] tensor T[i][2] then Cat::HopfAlgebraWithBasis returns the sum of the products T[i][1] * T[i][2]Cat::HopfAlgebraWithBasis – the category of Hopf algebras over a distinguished basis
Cat::HopfAlgebraWithBasis represents the category of Hopf algebras over a distinguished basis.
Creating the Category
Cat::HopfAlgebraWithBasis(R)
Parameters:
|
R: |
A domain which must be from the category Cat::Ring. |
Categories
Details:
A Cat::HopfAlgebraWithBasis is a Hopf algebra over a distinguished basis. The Hopf algebra operations (product, coproduct, counit) are typically implemented by extending by linearity their definitions on the basis elements.
Entries
|
"tensorSquare" |
The domain of the tensor products of the Hopf algebra with itself. |
Mathematical Methods
mu – The linearization of the product.
Cat::HopfAlgebraWithBasis::mu(dom::tensorSquare tx)
Returns the linearized product of tx, that is if tx is the sum over of T[i][1] tensor T[i][2] then Cat::HopfAlgebraWithBasis returns the sum of the products T[i][1] * T[i][2]
coproduct – The coproduct.
Cat::HopfAlgebraWithBasis::coproduct(dom x)
Returns the coproduct of x.
If an entry dom::coproductBasis is provided, then Cat::HopfAlgebraWithBasisis defined by extending it by linearity.
coproductBasis – The coproduct on basis elements.
Cat::HopfAlgebraWithBasis::coproductBasis(dom::basisIndices ind)
Returns the coproduct of dom::term(ind), the basis elements indexed by ind. If this methods is provided, it is used to define by linear extension the method dom::coproduct.
counit – The counit.
Cat::HopfAlgebraWithBasis::counit(dom x)
Returns the counit of x.
antipode – The coproduct.
Cat::HopfAlgebraWithBasis::antipode(dom x)
Returns the antipode of x.
If an entry dom::antipodeBasis is provided, then Cat::HopfAlgebraWithBasisis defined by extending it by linearity.
antipodeBasis – The antipode on basis elements.
Cat::HopfAlgebraWithBasis::antipodeBasis(dom::basisIndices ind)
Returns the antipode of dom::term(ind), the basis elements indexed by ind. If this methods is provided, it is used to define by linear extension the method dom::antipode.
tensorConstructor – The constructor for tensor elements.
Cat::HopfAlgebraWithBasis::tensorConstructor(dom x, dom y)
Returns the tensor product of x and y.
toCartesianProduct – Convert for tensor products into Cartesian products.
Cat::HopfAlgebraWithBasis::toCartesianProduct(dom::tensorSquare tx)
Returns a lists of pairs of elements of dom whose sum of tensor products is equal to tx.