Cat::GradedHopfAlgebraWithBasis – the category of graded Hopf algebras over a distinguished basis

Cat::GradedHopfAlgebraWithBasis represents the category of graded Hopf algebras over a distinguished basis.

Creating the Category

Cat::GradedHopfAlgebraWithBasis(R)

Categories

Cat::HopfAlgebraWithBasis(R), Cat::GradedAlgebraWithBasis(R)

Details:

• A Cat::GradedHopfAlgebraWithBasis is an Hopf algebra over a distinguished basis (Cat::AlgebraWithBasis), with a grading which is compatible with the Hopf algebra operations.

• Currently we assume the that the Hopf algebra is connected, that is the homogeneous component of degree zero is one dimensional and is identified with the scalars. Thus the counit can be automatically computed as well as the antipode, the later in a very slow manner.

counit – The counit.

Cat::GradedHopfAlgebraWithBasis::counit(dom x)

Returns the counit of x, that is the coefficient of the zero-th degree basis element.

antipodeBasis – The antipode on basis elements.

Cat::GradedHopfAlgebraWithBasis::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.

If not overloaded, the computation is done by inverting the identity using , which can be very slow.