#16508: Add Commutative graded differential algebras.
-------------------------------------+-------------------------------------
Reporter: mmarco | Owner:
Type: enhancement | Status: needs_review
Priority: major | Milestone: sage-6.3
Component: algebra | Resolution:
Keywords: sd58, sd59, | Merged in:
algebras, nonconmutative, graded | Reviewers:
Authors: mmarco | Work issues:
Report Upstream: N/A | Commit:
Branch: | f5f4c1d8ab67d5ca09046e74aa4544140a58e86e
u/jhpalmieri/DGA_new | Stopgaps:
Dependencies: |
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Description changed by jhpalmieri:
Old description:
> This patch adds basic CDGA's.
>
> They work as follows, first you create the algebra:
>
> {{{
> sage: A = CDGAlgebra(QQ, 'x,y,t', degrees = (1, 1, 2))
> sage: A
> Commutative Graded Differential Algebra over Rational Field with
> generators ('x', 'y', 't')
> sage: A.inject_variables()
> Defining x, y, t
> }}}
>
> Then define the differential:
> {{{
> sage: D = A.differential({x: x*y , y: x*y })
> sage: D
> Differential map in Commutative Graded Differential Algebra over Rational
> Field with generators ('x', 'y', 't')
> sending:
> x --> x*y
> y --> x*y
> t --> 0
>
> }}}
>
> Now you can compute things like a basis of each homogeneous part:
>
> {{{
> sage: A.homogeneous_part(5)
> [y*t^2, x*t^2]
> }}}
>
> Or the cohomology at each degree:
>
> {{{
> sage: A.cohomology(4, D)
> Vector space quotient V/W of dimension 1 over Rational Field where
> V: Vector space of degree 2 and dimension 2 over Rational Field
> Basis matrix:
> [1 0]
> [0 1]
> W: Vector space of degree 2 and dimension 1 over Rational Field
> Basis matrix:
> [0 1]
> }}}
New description:
This patch adds basic CDGA's.
They work as follows, first you create the algebra:
{{{
sage: A.<x,y,t> = CDGAlgebra(QQ, degrees = (1, 1, 2))
sage: A
Graded Commutative Algebra over Rational Field with generators ('x', 'y',
't') in degrees (1, 1, 2)
}}}
Then define the differential on `A` to construct a differential graded
algebra:
{{{
sage: sage: B = A.CDGAlgebra({x: x*y , y: x*y })
sage: B
Commutative Differential Graded Algebra over Rational Field with
generators ('x', 'y', 't') in degrees (1, 1, 2), differential
x --> x*y
y --> x*y
t --> 0
}}}
Now you can compute things like a basis of each homogeneous part:
{{{
sage: B.basis(5)
[y*t^2, x*t^2]
}}}
Or the cohomology in each degree:
{{{
sage: B.cohomology(4)
Free module generated by {[t^2]} over Rational Field
sage: B.cohomology(9)
Free module generated by {[-x*t^4 + y*t^4]} over Rational Field
}}}
--
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Ticket URL: <http://trac.sagemath.org/ticket/16508#comment:28>
Sage <http://www.sagemath.org>
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