ASKSAGE: Sage Q&A Forum - Latest question feedhttp://ask.sagemath.org/questions/Q&A Forum for SageenCopyright Sage, 2010. Some rights reserved under creative commons license.Thu, 22 Jun 2017 15:03:06 -0500How do I find the image of an element of a differential algebra in the cohomlogy?http://ask.sagemath.org/question/38060/how-do-i-find-the-image-of-an-element-of-a-differential-algebra-in-the-cohomlogy/Following the documentation on Commutative Differential Graded Algebras, I have defined a differential graded algebra $C$. I have some element $x \in C$, in degree $4$. I can get a basis for the cohomology at degree 4 by
C.cohomology(4)
and generators for cocycles and coboundaries by
C.cocycles(4)
C.coboundaries(4)
How do I check if $x$ is a cocycle, and if it is, what it is in terms of the basis of the cohomology above?
I'm not sure I used the right tags, feel free to edit.ronnoThu, 22 Jun 2017 15:03:06 -0500http://ask.sagemath.org/question/38060/How to make a vector space with basis a *graded* one, and how to do linear algebra on its homogeneous parts?http://ask.sagemath.org/question/8671/how-to-make-a-vector-space-with-basis-a-graded-one-and-how-to-do-linear-algebra-on-its-homogeneous-parts/I was sent here from [StackOverflow](http://stackoverflow.com/questions/9018933).
I have a vector space with given basis (it is also a Hopf algebra, but this is not part of the problem). How do I make it into a graded vector space? E. g., I know that in order to make it into an algebra, I have to define a function called `product_on_basis` somewhere in its definition, and that in order to make it into a coalgebra, I have to define a function called `coproduct_on_basis`; but what function do I have to define in order to make it into a graded vector space? How can I find out the name of this function? (It is not given in http://www.sagemath.org/doc/reference/sage/categories/graded_modules_with_basis.html . I know the names of the functions for the multiplication and the comultiplication from python2.6/site-packages/sage/categories/examples/hopf_algebras_with_basis.py , but I don't see such a .py file for graded vector spaces.)
Once this is done, I would like to do linear algebra on the graded components. They are each finite-dimensional, with basis a part of the combinatorial basis of the big space, so there shouldn't be any problem. I have defined two maps and want to know, e. g., whether the image of one lies inside the image of the other. Is there an abstract way to do this in Sage or do I have to translate these maps into matrices?
**Context (not important):** I have (successfully, albeit stupidly) implemented the Malvenuto-Reutenauer Hopf algebra of permutations:
[html version][1] resp. [sws file][2]
Now I want to check [some of its properties][3]. This checking cannot be automated on the *whole* space, but it is a finite problem on each of its graded components, so I would like to check it, say, on the fifth one.
[1]: http://mit.edu/~darij/www/mrh.htm
[2]: http://mit.edu/~darij/www/mrh.sws
[3]: http://mathoverflow.net/questions/84345/darijgrinbergThu, 26 Jan 2012 06:12:18 -0600http://ask.sagemath.org/question/8671/How can I construct graded algebras?http://ask.sagemath.org/question/8622/how-can-i-construct-graded-algebras/I am trying to create a graded algebra using generators and relations. I found that sage has a category for such things:
[http://www.sagemath.org/doc/reference/sage/categories/graded_modules_with_basis.html](http://www.sagemath.org/doc/reference/sage/categories/graded_modules_with_basis.html)
but there are no constructors or examples of how to create these things. Does anyone know where I can find examples of how to construct graded algebras, or more generally how to construct non-commutative algebras?StarxThu, 12 Jan 2012 12:49:07 -0600http://ask.sagemath.org/question/8622/