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3 Graded algebras
3.1 Graded algebras in HAP (and HAPprime)
3.2 Data access functions
3.2-1 ModPRingGeneratorDegrees
3.2-2 ModPRingNiceBasis
3.2-3 ModPRingNiceBasisAsPolynomials
3.2-4 ModPRingBasisAsPolynomials
3.3 Other functions
3.3-1 PresentationOfGradedStructureConstantAlgebra
3.4 Example: Graded algebras and mod-p cohomology rings

### 3 Graded algebras

A graded algebra A is a ring that has a direct sum decomposition into Abelian additive groups (called degrees)

A = A_1 + A_2 + A_3 + ...

and where multiplication of elements follows a grading such that

A_s x A_r --> A_s+r

This means that for any elements x in A_s and y in A_r, the product lies in degree r+s, i.e. xy in A_s+r.

#### 3.1 Graded algebras in HAP (and HAPprime)

Small finite algebras can be represented explicitly in GAP using a basis and a structure constant table Tutorial: Algebras, which specifies all products of basis elements. HAP extends this definition to provide a finite approximation to a graded algebra, storing all basis elements and products up to a given degree, n. The graded algebras used by HAP add a new component to the algebra object `A`, the function `A!.degree(e)` which returns the degree of an algebra element `e`.

HAPprime uses this HAP algebra object and adds some attributes, providing information on the basis and generators of the algebra. A ring presentation for this algebra can also be computed and represented using the `GradedAlgebraPresentation` datatype (see Section 4).

#### 3.2 Data access functions

##### 3.2-1 ModPRingGeneratorDegrees

`> ModPRingGeneratorDegrees`( A ) | ( attribute )

Returns: List

Returns a list containing the degree of each generator in a minimal generating set for the mod-p cohomology ring A. The ith degree in the list corresponds to the ith generator returned by `ModPRingGenerators` (HAP: ModPRingGenerators)

##### 3.2-2 ModPRingNiceBasis

`> ModPRingNiceBasis`( A ) | ( attribute )

Returns: List

Returns the information needed to convert the basis for A (see `Basis` (Reference: Basis)) into a nicer basis consisting of only products of ring generators. The function returns a pair of lists `[Coeff, Bas]`. The list `Coeff` is a change-of-basis matrix, where the ith row gives the standard basis element i in terms of the nice basis. The list `Bas` can be used to form the new basis and is a list of integers where the ith 'nice basis' element is given by `Product(List(Bas[i], x->Basis(A)[x]))`.

This attribute returns exactly the same list as is provided by component `A!.niceBasis` (see `ModPCohomologyRing` (HAP: ModPCohomologyRing), but automatically constructs this list if it is not available.

##### 3.2-3 ModPRingNiceBasisAsPolynomials

`> ModPRingNiceBasisAsPolynomials`( A ) | ( attribute )

Returns: List

A list which gives the 'nice basis' of the algebra A (as returned by the second element of `ModPRingNiceBasis` (3.2-2)) in terms of products of the indeterminates in the ring presentation (as given by `PresentationOfGradedStructureConstantAlgebra` (3.3-1)). The ith entry in the list corresponds to the ith basis element returned by `ModPRingNiceBasis` (3.2-2).

##### 3.2-4 ModPRingBasisAsPolynomials

`> ModPRingBasisAsPolynomials`( A ) | ( attribute )

Returns: List

A list which gives the basis of the algebra A (as returned by `Basis(A)`) in terms of sums of products of the indeterminates in the ring presentation (as given by `PresentationOfGradedStructureConstantAlgebra` (3.3-1)). The ith entry in the list corresponds to the ith basis element returned by `Basis` (Reference: Basis).

#### 3.3 Other functions

##### 3.3-1 PresentationOfGradedStructureConstantAlgebra

`> PresentationOfGradedStructureConstantAlgebra`( A ) | ( attribute )

Returns: `GradedAlgebraPresentation`

Returns a ring presentation for the graded algebra A. The ring A must be a structure constant algebra with embedded degrees, such as is returned by `ModPCohomologyRing` (HAP: ModPCohomologyRing). The generators of the `GradedAlgebraPresentation` (as returned by `IndeterminatesOfGradedAlgebraPresentation` (4.3-3) are in one-to-one correspondance with the generators of A as returned by `ModPRingGenerators` (HAP: ModPRingGenerators) (ignoring the first generator, which is in degree zero).

#### 3.4 Example: Graded algebras and mod-p cohomology rings

The mod-p cohomology ring of a p-group is a graded algebra (see HAPprime: Computing mod-p cohomology rings and their Poincaré series). As an example, we use the HAP function `ModPCohomologyRing` (HAP: ModPCohomologyRing) to compute this algebra for the quaternion group Q_8 up to and including degree five. We can display its generators and basis (and degrees), and the information to construct a 'nice basis'

    gap> A := ModPCohomologyRing(SmallGroup(8, 4), 5);
    <algebra of dimension 9 over GF(2)>
    gap> Display(Basis(A));
    CanonicalBasis( Algebra( GF(2), [ v.1, v.2, v.3, v.4, v.5, v.6, v.7, v.8, v.9
     ] ) )
    gap> List(Basis(A), i->A!.degree(i));
    [ 0, 1, 1, 2, 2, 3, 4, 5, 5 ]
    gap> ModPRingGenerators(A);
    [ v.1, v.2, v.3, v.7 ]
    gap> ModPRingGeneratorDegrees(A);
    [ 0, 1, 1, 4 ]
    gap> ModPRingNiceBasis(A);
    [ [ [ 1, 0, 0, 0, 0, 0, 0, 0, 0 ], [ 0, 1, 0, 0, 0, 0, 0, 0, 0 ],
          [ 0, 0, 1, 0, 0, 0, 0, 0, 0 ], [ 0, 0, 0, 0, 0, 1, 0, 0, 0 ],
          [ 0, 0, 0, 0, 1, 1, 0, 0, 0 ], [ 0, 0, 0, 0, 0, 0, 0, 0, 1 ],
          [ 0, 0, 0, 1, 0, 0, 0, 0, 0 ], [ 0, 0, 0, 0, 0, 0, 0, 1, 0 ],
          [ 0, 0, 0, 0, 0, 0, 1, 1, 0 ] ],
      [ [ 1, 1, 1 ], [ 1, 1, 2 ], [ 1, 1, 3 ], [ 1, 1, 7 ], [ 1, 2, 2 ],
          [ 1, 2, 3 ], [ 1, 2, 7 ], [ 1, 3, 7 ], [ 2, 2, 3 ] ] ]

A presentation for this algebra can be constructed using `PresentationOfGradedStructureConstantAlgebra` (3.3-1), which (see below) returns the presentation to be

H^*(G, F) = F[x_1, x_2, x_3] / (x_1^2 + x_1x_2 + x_2^2, x_2^3)

where x_1 and x_2 have degree one and x_3 is degree four. The 'nice basis' referred to above is monomials of degree up to five in this polynomial ring, and is not necessarily the same as the basis of the algebra given by `Basis` (Reference: Basis), as demonstrated below.

    gap> A := ModPCohomologyRing(SmallGroup(8, 4), 5);;
    gap> PresentationOfGradedStructureConstantAlgebra(A);
    Graded algebra GF(2)[ x_1, x_2, x_3 ] / [ x_1^2+x_1*x_2+x_2^2, x_2^3
     ] with indeterminate degrees [ 1, 1, 4 ]

    gap> ModPRingNiceBasisAsPolynomials(A);
    [ Z(2)^0, x_1, x_2, x_3, x_1^2, x_1*x_2, x_1*x_3, x_2*x_3, x_1^2*x_2 ]
    gap> ModPRingBasisAsPolynomials(A);
    [ Z(2)^0, x_1, x_2, x_1*x_2, x_1^2+x_1*x_2, x_1^2*x_2, x_3, x_2*x_3,
      x_1*x_3+x_2*x_3 ]

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