Non-homogenous Ideals In Unital Associative Free Algebras

[Abdolrasool Bahari-fard]

Hi all,

I want to define a two sided ideal I=[x*y*x*y-x*y, y*x*y*x-x*y*x] in an unital associative free algebra F.<x,y>. (not just in a free algebra)

I wrote in sage:

F.<x,y>=FreeAlgebra(QQ, implementation='letterplace')
I=F*[x*y*x*y-x*y, y*x*y*x-x*y*x]*F
I()

"But, the error I have faced to is the following:

ArithmaticError: Can only subtract the elements of the same degree."

Also I want to know how can I create a non-commutative infinite dimensional quotient algebra F/I.

Is there any way to define such an ideal and this quotient in sage or other part of sage like Gap or Singular?

Thanks

[Simon King]

Hi!

Am Dienstag, 8. Oktober 2013 18:04:22 UTC+2 schrieb Abdolrasool Bahari-fard:

 want to define a two sided ideal I=[x*y*x*y-x*y, y*x*y*x-x*y*x] in an unital associative free algebra F.<x,y>. (not just in a free algebra)

I wrote in sage:

F.<x,y>=FreeAlgebra(QQ, implementation='letterplace')
I=F*[x*y*x*y-x*y, y*x*y*x-x*y*x]*F
I()

"But, the error I have faced to is the following:

ArithmeticError: Can only subtract the elements of the same degree."

This error only occurs when working with Letterplace. With the default implementation, it is no problem to define the ideal:

sage: F.<x,y> = FreeAlgebra(QQ)
sage: I = F*[x*y*x*y-y*x]*F
sage: I
Twosided Ideal (-y*x + x*y*x*y) of Free Algebra on 2 generators (x, y) over Rational Field

However, the default implementation does not know about computation of Gröbner bases in free associative algebras. For this, you'd need the implementation that is based on Letterplace.

If I am not mistaken, the developers of Letterplace in Singular currently try to make the non-homogeneous case work, but this is not done, at least not with the version of Singular that is shipped as part of Sage. You can find the documentation of Letterplace on this and on subsequent pages of the Singular manual: http://www.singular.uni-kl.de/Manual/latest/sing_534.htm#SEC586
 
You will find that it is rather awkward to directly work with Letterplace in Singular. The wrapper in Sage is more comfortable. You may try the non-homogeneous case in Singular, but when I wrote the wrapper in Sage, I found that the non-homogeneous case was extremely buggy, and therefore I left it out.

Is there any way to define such an ideal and this quotient in sage or other part of sage like Gap or Singular?

I don't know what exactly Gap can do, but cited from the manual at http://www.gap-system.org/Manuals/doc/ref/chap63.html#X85A22A8286596D02:

"Currently the GAP library contains only few functions dealing with general finitely presented algebras, so this chapter is merely a placeholder. The special case of finitely presented Lie algebras is described in 64.11, and there is also a GAP package fplsa for computing structure constants of finitely presented Lie (super)algebras."

Concerning Singular, even if you get letterplace to work in the non-homogeneous case, you still could not easily work with quotient algebras, since this would require computing normal forms with respect to the relation ideal --- and currently this is not provided by Letterplace in Singular. But I implemented the computation of normal forms in the wrapper for Letterplace in Sage, so, perhaps you can see by reading the code how it could in principle be done.

Your example should be doable somehow, but of course general finitely presented algebras are computationally intractable.

Best regards,

Simon