Filtered Algebras With Basis#
A filtered algebra with basis over a commutative ring \(R\)
is a filtered algebra over \(R\) endowed with the structure
of a filtered module with basis (with the same underlying
filtered-module structure). See
FilteredAlgebras
and
FilteredModulesWithBasis
for these two notions.
- class sage.categories.filtered_algebras_with_basis.FilteredAlgebrasWithBasis(base_category)#
Bases:
FilteredModulesCategory
The category of filtered algebras with a distinguished homogeneous basis.
A filtered algebra with basis over a commutative ring \(R\) is a filtered algebra over \(R\) endowed with the structure of a filtered module with basis (with the same underlying filtered-module structure). See
FilteredAlgebras
andFilteredModulesWithBasis
for these two notions.EXAMPLES:
sage: C = AlgebrasWithBasis(ZZ).Filtered(); C Category of filtered algebras with basis over Integer Ring sage: sorted(C.super_categories(), key=str) [Category of algebras with basis over Integer Ring, Category of filtered algebras over Integer Ring, Category of filtered modules with basis over Integer Ring]
- class ElementMethods#
Bases:
object
- class ParentMethods#
Bases:
object
- from_graded_conversion()#
Return the inverse of the canonical \(R\)-module isomorphism \(A \to \operatorname{gr} A\) induced by the basis of \(A\) (where \(A = ` ``self`\)). This inverse is an isomorphism \(\operatorname{gr} A \to A\).
This is an isomorphism of \(R\)-modules, not of algebras. See the class documentation
AssociatedGradedAlgebra
.See also
EXAMPLES:
sage: A = Algebras(QQ).WithBasis().Filtered().example() sage: p = A.an_element() + A.algebra_generators()['x'] + 2; p U['x']^2*U['y']^2*U['z']^3 + 3*U['x'] + 3*U['y'] + 3 sage: q = A.to_graded_conversion()(p) sage: A.from_graded_conversion()(q) == p True sage: q.parent() is A.graded_algebra() True
- graded_algebra()#
Return the associated graded algebra to
self
.See
AssociatedGradedAlgebra
for the definition and the properties of this.If the filtered algebra
self
with basis is called \(A\), then this method returns \(\operatorname{gr} A\). The methodto_graded_conversion()
returns the canonical \(R\)-module isomorphism \(A \to \operatorname{gr} A\) induced by the basis of \(A\), and the methodfrom_graded_conversion()
returns the inverse of this isomorphism. The methodprojection()
projects elements of \(A\) onto \(\operatorname{gr} A\) according to their place in the filtration on \(A\).Warning
When not overridden, this method returns the default implementation of an associated graded algebra – namely,
AssociatedGradedAlgebra(self)
, whereAssociatedGradedAlgebra
isAssociatedGradedAlgebra
. But many instances ofFilteredAlgebrasWithBasis
override this method, as the associated graded algebra often is (isomorphic) to a simpler object (for instance, the associated graded algebra of a graded algebra can be identified with the graded algebra itself). Generic code that uses associated graded algebras (such as the code of theinduced_graded_map()
method below) should make sure to only communicate with them via theto_graded_conversion()
,from_graded_conversion()
, andprojection()
methods (in particular, do not expect there to be a conversion fromself
toself.graded_algebra()
; this currently does not work for Clifford algebras). Similarly, when overridinggraded_algebra()
, make sure to accordingly redefine these three methods, unless their definitions below still apply to your case (this will happen whenever the basis of yourgraded_algebra()
has the same indexing set asself
, and the partition of this indexing set according to degree is the same as forself
).Todo
Maybe the thing about the conversion from
self
toself.graded_algebra()
on the Clifford at least could be made to work? (I would still warn the user against ASSUMING that it must work – as there is probably no way to guarantee it in all cases, and we shouldn’t require users to mess with element constructors.)EXAMPLES:
sage: A = AlgebrasWithBasis(ZZ).Filtered().example() sage: A.graded_algebra() Graded Algebra of An example of a filtered algebra with basis: the universal enveloping algebra of Lie algebra of RR^3 with cross product over Integer Ring
- induced_graded_map(other, f)#
Return the graded linear map between the associated graded algebras of
self
andother
canonically induced by the filtration-preserving mapf : self -> other
.Let \(A\) and \(B\) be two filtered algebras with basis, and let \((F_i)_{i \in I}\) and \((G_i)_{i \in I}\) be their filtrations. Let \(f : A \to B\) be a linear map which preserves the filtration (i.e., satisfies \(f(F_i) \subseteq G_i\) for all \(i \in I\)). Then, there is a canonically defined graded linear map \(\operatorname{gr} f : \operatorname{gr} A \to \operatorname{gr} B\) which satisfies
\[(\operatorname{gr} f) (p_i(a)) = p_i(f(a)) \qquad \text{for all } i \in I \text{ and } a \in F_i ,\]where the \(p_i\) on the left hand side is the canonical projection from \(F_i\) onto the \(i\)-th graded component of \(\operatorname{gr} A\), while the \(p_i\) on the right hand side is the canonical projection from \(G_i\) onto the \(i\)-th graded component of \(\operatorname{gr} B\).
INPUT:
other
– a filtered algebra with basisf
– a filtration-preserving linear map fromself
toother
(can be given as a morphism or as a function)
OUTPUT:
The graded linear map \(\operatorname{gr} f\).
EXAMPLES:
Example 1.
We start with the universal enveloping algebra of the Lie algebra \(\RR^3\) (with the cross product serving as Lie bracket):
sage: A = AlgebrasWithBasis(QQ).Filtered().example(); A An example of a filtered algebra with basis: the universal enveloping algebra of Lie algebra of RR^3 with cross product over Rational Field sage: M = A.indices(); M Free abelian monoid indexed by {'x', 'y', 'z'} sage: x,y,z = [A.basis()[M.gens()[i]] for i in "xyz"]
Let us define a stupid filtered map from
A
to itself:sage: def map_on_basis(m): ....: d = m.dict() ....: i = d.get('x', 0); j = d.get('y', 0); k = d.get('z', 0) ....: g = (y ** (i+j)) * (z ** k) ....: if i > 0: ....: g += i * (x ** (i-1)) * (y ** j) * (z ** k) ....: return g sage: f = A.module_morphism(on_basis=map_on_basis, ....: codomain=A) sage: f(x) U['y'] + 1 sage: f(x*y*z) U['y']^2*U['z'] + U['y']*U['z'] sage: f(x*x*y*z) U['y']^3*U['z'] + 2*U['x']*U['y']*U['z'] sage: f(A.one()) 1 sage: f(y*z) U['y']*U['z']
(There is nothing here that is peculiar to this universal enveloping algebra; we are only using its module structure, and we could just as well be using a polynomial algebra in its stead.)
We now compute \(\operatorname{gr} f\)
sage: grA = A.graded_algebra(); grA Graded Algebra of An example of a filtered algebra with basis: the universal enveloping algebra of Lie algebra of RR^3 with cross product over Rational Field sage: xx, yy, zz = [A.to_graded_conversion()(i) for i in [x, y, z]] sage: xx+yy*zz bar(U['y']*U['z']) + bar(U['x']) sage: grf = A.induced_graded_map(A, f); grf Generic endomorphism of Graded Algebra of An example of a filtered algebra with basis: the universal enveloping algebra of Lie algebra of RR^3 with cross product over Rational Field sage: grf(xx) bar(U['y']) sage: grf(xx*yy*zz) bar(U['y']^2*U['z']) sage: grf(xx*xx*yy*zz) bar(U['y']^3*U['z']) sage: grf(grA.one()) bar(1) sage: grf(yy*zz) bar(U['y']*U['z']) sage: grf(yy*zz-2*yy) bar(U['y']*U['z']) - 2*bar(U['y'])
Example 2.
We shall now construct \(\operatorname{gr} f\) for a different map \(f\) out of the same
A
; the new map \(f\) will lead into a graded algebra already, namely into the algebra of symmetric functions:sage: # needs sage.combinat sage.modules sage: h = SymmetricFunctions(QQ).h() sage: def map_on_basis(m): # redefining map_on_basis ....: d = m.dict() ....: i = d.get('x', 0); j = d.get('y', 0); k = d.get('z', 0) ....: g = (h[1] ** i) * (h[2] ** (j // 2) * (h[3] ** (k // 3))) ....: g += i * (h[1] ** (i+j+k)) ....: return g sage: f = A.module_morphism(on_basis=map_on_basis, ....: codomain=h) # redefining f sage: f(x) 2*h[1] sage: f(y) h[] sage: f(z) h[] sage: f(y**2) h[2] sage: f(x**2) 3*h[1, 1] sage: f(x*y*z) h[1] + h[1, 1, 1] sage: f(x*x*y*y*z) 2*h[1, 1, 1, 1, 1] + h[2, 1, 1] sage: f(A.one()) h[]
The algebra
h
of symmetric functions in the \(h\)-basis is already graded, so its associated graded algebra is implemented as itself:sage: # needs sage.combinat sage.modules sage: grh = h.graded_algebra(); grh is h True sage: grf = A.induced_graded_map(h, f); grf Generic morphism: From: Graded Algebra of An example of a filtered algebra with basis: the universal enveloping algebra of Lie algebra of RR^3 with cross product over Rational Field To: Symmetric Functions over Rational Field in the homogeneous basis sage: grf(xx) 2*h[1] sage: grf(yy) 0 sage: grf(zz) 0 sage: grf(yy**2) h[2] sage: grf(xx**2) 3*h[1, 1] sage: grf(xx*yy*zz) h[1, 1, 1] sage: grf(xx*xx*yy*yy*zz) 2*h[1, 1, 1, 1, 1] sage: grf(grA.one()) h[]
Example 3.
After having had a graded algebra as the codomain, let us try to have one as the domain instead. Our new
f
will go fromh
toA
:sage: # needs sage.combinat sage.modules sage: def map_on_basis(lam): # redefining map_on_basis ....: return x ** (sum(lam)) + y ** (len(lam)) sage: f = h.module_morphism(on_basis=map_on_basis, ....: codomain=A) # redefining f sage: f(h[1]) U['x'] + U['y'] sage: f(h[2]) U['x']^2 + U['y'] sage: f(h[1, 1]) U['x']^2 + U['y']^2 sage: f(h[2, 2]) U['x']^4 + U['y']^2 sage: f(h[3, 2, 1]) U['x']^6 + U['y']^3 sage: f(h.one()) 2 sage: grf = h.induced_graded_map(A, f); grf Generic morphism: From: Symmetric Functions over Rational Field in the homogeneous basis To: Graded Algebra of An example of a filtered algebra with basis: the universal enveloping algebra of Lie algebra of RR^3 with cross product over Rational Field sage: grf(h[1]) bar(U['x']) + bar(U['y']) sage: grf(h[2]) bar(U['x']^2) sage: grf(h[1, 1]) bar(U['x']^2) + bar(U['y']^2) sage: grf(h[2, 2]) bar(U['x']^4) sage: grf(h[3, 2, 1]) bar(U['x']^6) sage: grf(h.one()) 2*bar(1)
Example 4.
The construct \(\operatorname{gr} f\) also makes sense when \(f\) is a filtration-preserving map between graded algebras.
sage: # needs sage.combinat sage.modules sage: def map_on_basis(lam): # redefining map_on_basis ....: return h[lam] + h[len(lam)] sage: f = h.module_morphism(on_basis=map_on_basis, ....: codomain=h) # redefining f sage: f(h[1]) 2*h[1] sage: f(h[2]) h[1] + h[2] sage: f(h[1, 1]) h[1, 1] + h[2] sage: f(h[2, 1]) h[2] + h[2, 1] sage: f(h.one()) 2*h[] sage: grf = h.induced_graded_map(h, f); grf Generic endomorphism of Symmetric Functions over Rational Field in the homogeneous basis sage: grf(h[1]) 2*h[1] sage: grf(h[2]) h[2] sage: grf(h[1, 1]) h[1, 1] + h[2] sage: grf(h[2, 1]) h[2, 1] sage: grf(h.one()) 2*h[]
Example 5.
For another example, let us compute \(\operatorname{gr} f\) for a map \(f\) between two Clifford algebras:
sage: # needs sage.modules sage: Q = QuadraticForm(ZZ, 2, [1,2,3]) sage: B = CliffordAlgebra(Q, names=['u','v']); B The Clifford algebra of the Quadratic form in 2 variables over Integer Ring with coefficients: [ 1 2 ] [ * 3 ] sage: m = Matrix(ZZ, [[1, 2], [1, -1]]) sage: f = B.lift_module_morphism(m, names=['x','y']) sage: A = f.domain(); A The Clifford algebra of the Quadratic form in 2 variables over Integer Ring with coefficients: [ 6 0 ] [ * 3 ] sage: x, y = A.gens() sage: f(x) u + v sage: f(y) 2*u - v sage: f(x**2) 6 sage: f(x*y) -3*u*v + 3 sage: grA = A.graded_algebra(); grA The exterior algebra of rank 2 over Integer Ring sage: A.to_graded_conversion()(x) x sage: A.to_graded_conversion()(y) y sage: A.to_graded_conversion()(x*y) x*y sage: u = A.to_graded_conversion()(x*y+1); u x*y + 1 sage: A.from_graded_conversion()(u) x*y + 1 sage: A.projection(2)(x*y+1) x*y sage: A.projection(1)(x+2*y-2) x + 2*y sage: grf = A.induced_graded_map(B, f); grf Generic morphism: From: The exterior algebra of rank 2 over Integer Ring To: The exterior algebra of rank 2 over Integer Ring sage: grf(A.to_graded_conversion()(x)) u + v sage: grf(A.to_graded_conversion()(y)) 2*u - v sage: grf(A.to_graded_conversion()(x**2)) 6 sage: grf(A.to_graded_conversion()(x*y)) -3*u*v sage: grf(grA.one()) 1
- projection(i)#
Return the \(i\)-th projection \(p_i : F_i \to G_i\) (in the notations of the class documentation
AssociatedGradedAlgebra
, where \(A = ` ``self`\)).This method actually does not return the map \(p_i\) itself, but an extension of \(p_i\) to the whole \(R\)-module \(A\). This extension is the composition of the \(R\)-module isomorphism \(A \to \operatorname{gr} A\) with the canonical projection of the graded \(R\)-module \(\operatorname{gr} A\) onto its \(i\)-th graded component \(G_i\). The codomain of this map is \(\operatorname{gr} A\), although its actual image is \(G_i\). The map \(p_i\) is obtained from this map by restricting its domain to \(F_i\) and its image to \(G_i\).
EXAMPLES:
sage: A = Algebras(QQ).WithBasis().Filtered().example() sage: p = A.an_element() + A.algebra_generators()['x'] + 2; p U['x']^2*U['y']^2*U['z']^3 + 3*U['x'] + 3*U['y'] + 3 sage: q = A.projection(7)(p); q bar(U['x']^2*U['y']^2*U['z']^3) sage: q.parent() is A.graded_algebra() True sage: A.projection(8)(p) 0
- to_graded_conversion()#
Return the canonical \(R\)-module isomorphism \(A \to \operatorname{gr} A\) induced by the basis of \(A\) (where \(A = ` ``self`\)).
This is an isomorphism of \(R\)-modules, not of algebras. See the class documentation
AssociatedGradedAlgebra
.See also
EXAMPLES:
sage: A = Algebras(QQ).WithBasis().Filtered().example() sage: p = A.an_element() + A.algebra_generators()['x'] + 2; p U['x']^2*U['y']^2*U['z']^3 + 3*U['x'] + 3*U['y'] + 3 sage: q = A.to_graded_conversion()(p); q bar(U['x']^2*U['y']^2*U['z']^3) + 3*bar(U['x']) + 3*bar(U['y']) + 3*bar(1) sage: q.parent() is A.graded_algebra() True