Generating a certain list of non-commuting polynomials with Sage
I have a set of variables $x_1,...,x_n$ and $y_1,...,y_m$ for $n,m >=1$.
Now I can build all quadratic monomials of the form $x_i y_j$ and $y_j x_i$ (but we do not have $x_i y_j= y_j x_i$ as we calcualte in the non-commutative polynomial ring). But something like $x_i x_j$ is not allowed as after an $x_i$ there must come an $y_j$ and after an $y_i$ there must come an $x_i$.
Now I want with Sage the list of all possible relations of the form $w_1 \pm w_2 \pm w_3 \cdots$ such that all $w_i$ are different quadratic relations that all start either with a $x_i$ or a $y_j$.
For example for $n=2$ and $m=1$, possible relations are (I hope I did not forget any relation) : $x_1 y_1, x_1 y_1-x_2 y_2, x_1 y_2 + x_2 y_2 , x_2 y_1,y_1 x_1, y_1 x_2, y_1 x_1 - y_1 x_2 , y_1x_1+y_1 x_2$.
I am not sure how to do this in an easy way with Sage, but maybe someone knows a simple trick.
Thanks for any help.
Please provide a code defining $x_i$ and $y_j$.
I omitted a formal definition because it is a bit complicated. A formal definition would be to take the connected quiver $Q$ with two points 1 and 2 and arrows $x_i$ from 1 to 2 and arrows $y_i$ from 2 to 1. Then one would have restrictions in the post. One can also work over a non-commutative polynomial ring with the relations $x_i x_j =0$ and $y_i y_j=0$. I am not sure whether there is code to work with quivers like this in sage.
For this purpose one could also define $xi$ and $yj$ just as strings probably or formal non-commutative variables in sage to get the needed output. I might use the output mainly to continue to work in GAP, where quivers are available.
I understand that you want to work in a ring $R$ where the addition is commutative biut the multiplication is not ; that is :
$\forall x,\,y \in R^2,\ x+y=y+x$
$\exists x,\,y \in R^2,\ x\cdot y\neq y\cdot x$
Is that right ?
Also : do you consider quadratic monomials such as $x^2,\, x\in R$ ?
Your example lists only moinomials and binomials, whereas your definition would also accept trinomials such as $x_1y_1 + x_2y_1 + y_1x_1$ (etc...) and quadrinomials such as $x_1y_1 + x_2y_1 + y_1x_1+ y_1x_2$ (etc).
Could you either :
clarify your definition, or
complete your example ?
Thanks !