Quotient of free algebra on 2 generators (x, y) over rational field by a non-homogenous ideal

[Abdolrasool Bahari-fard]

Hi all,

Here I asked a question in sage but there is an error which I can not solve it:

F.<x,y>=FreeAlgebra(QQ)

I=F*[x*y*x*y-y*x, y*x*y*x-x*y]*F

G.<a,b>=F.quo(I)

G

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TypeError: quotient() takes exactly 4 arguments (3 given)

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Please help me to fine the forth argument should I put.

Actually I want to construct a quotient of infinite dimensional non-commutative free algebra F by a non-homogenous ideal.

Thanks

[Simon King]

Hi!

Am Mittwoch, 9. Oktober 2013 00:18:21 UTC+2 schrieb Abdolrasool Bahari-fard:

Here I asked a question in sage but there is an error which I can not solve it:

F.<x,y>=FreeAlgebra(QQ)

I=F*[x*y*x*y-y*x, y*x*y*x-x*y]*F

G.<a,b>=F.quo(I)

________________________________________________

TypeError: quotient() takes exactly 4 arguments (3 given)

_________________________________________________

Please help me to fine the forth argument should I put.

You can easily read the documentation of the "quo" function interactively: As usual in Python, put a question mark after the object you want to study (here: F.quo) and hit return (or shift-return in the notebook). You'll get:

sage: F.quo?
 Type:       instancemethod
String Form:<bound method FreeAlgebra_generic_with_category.quotient of Free Algebra on 2 generators (x, y) over Rational Field>
File:       /home/king/Sage/git/sage/local/lib/python2.7/site-packages/sage/algebras/free_algebra.py
Definition: F.quo(self, mons, mats, names)
Docstring:
   Returns a quotient algebra.

   The quotient algebra is defined via the action of a free algebra A
   on a (finitely generated) free module. The input for the quotient
   algebra is a list of monomials (in the underlying monoid for A)
   which form a free basis for the module of A, and a list of
   matrices, which give the action of the free generators of A on this
   monomial basis.

   EXAMPLE:
   ...
 
If you put two question marks (F.quo??) you'll even see the complete source code of the quo-method. And of course, Sage also has a reference manual that ought to be consulted.

Actually I want to construct a quotient of infinite dimensional non-commutative free algebra F by a non-homogenous ideal.

 
As you can see in the documentation, the method can not work with a relation ideal, but needs multiplication matrices. In particular, the quo-method of free algebras in default implementation is only dedicated to the case of finite dimensional quotients. Actually I think that the documentation should state this fact more clearly.

The letterplace implementation of free algebras has a different quo-method and does accept a relation ideal and does support infinite dimensional quotients. However, the relation ideal needs to be homogeneous in this case. Sorry.

Best regards,
Simon