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Creating a NC Algebra
https://www.singular.uni-kl.de/forum/viewtopic.php?f=10&t=1641		 Page 1 of 1
Author:  Justin [ Thu May 31, 2007 10:08 pm ]
Post subject:  Creating a NC Algebra
Hi,

I've tried out the examples in the Plural doc, and looked at
some postings in the Forum, but I am still unclear on how to
do the following.

I want an algebra, A, with generators 1,x,y,..., subject to relations
like this:
f(x)=0
g(y)=0
(so each generator generates a number field, assuming the base
field is Q).

Then there are relations like xy + yx + ... = 0.

The latter seems clear from the examples. I am having problems
with the former. I would think that creating a base qring using an
ideal generated by f,g,... would be the way to start, but the results
aren't what I expect (e.g., I always get a reduced value of 0 for x^2).

What am I missing?

Thanks!

Justin
Author:  levandov [ Tue Jun 12, 2007 7:45 pm ]
Post subject:  Re: Creating a NC Algebra
Hi Justin,

basically you're given say the generators x,y over some field K
and three relations, xy + yx + ... = 0, f(x)=0, g(y) = 0.
The classical way to input such an algebra is
1) define the algebra, say A, in x,y with the relation xy + yx + ... = 0 (with, say, ncalg command)
2) compute a two-sided Groebner basis of the ideal in A,
generated by polynomials f(x), g(y):
Code:
poly f = ...; poly g = ...;
ideal Q = f,g;
Q = twostd(Q);

Here it would be interesting to see what Q gives
3) pass to the factor algebra modulo Q
Code:
qring q = Q;


If you have further problems, please send me some examples you're playing with to levandov at mathematik.uni-kl.de

Best regards.
Viktor
Author:  Justin [ Wed Jun 20, 2007 6:23 pm ]
Post subject:  Re: Creating a NC Algebra
Thanks to some off-line help from Viktor, and a course in remedial reading :?, I have some success.

The problem I reported is due to the fact that I misread the example of an ncalgebra computation in the manual. Once I got that right, I was able to get sensible results.

For the record, the "nc" relation I was using is xy + yx + x + 1. I filled in the two arrays with
Code:
C[2][2] = 1
D[2][2] = x + 1

which, unfortunately, represents yx - xy - x - 1. As usual, the computer did exactly what I told it to.

Using the latter relation, it was easy to see that the ideal generated by the algebraic relations for x, y is in fact the unit ideal. This accounts for the fact that in the quotient ring, everything came up as 0.

Justin
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