# Ideals in the Group Algebra C[GL(2,R)]

I was hoping to use Sage to determine whether some element is generated by given generators in the group algebra C[GL(2,R)]. However, it went completely wrong when I tried the following naive thing.

sage: G=GL(2,RR)
sage: R=GroupAlgebra(G,CC)
sage: T=R(G([[1,1],[0,1]]))
sage: I=[T-1]*R;
sage: J=[T-1,(T-1)^2]*R;
sage: I==J
False


I thought I defined the ideals in the wrong way so I tried the following as well, which worked out great.

sage: var('t')
sage: R=PolynomialRing(QQ, 't')
sage: I=[R(t)]*R
sage: J=[R(t),R(t^2)]*R;
sage: I==J
True


I was wondering whether I indeed did something wrong or this is just a bug of Sage, and whether there is a way to fix it or bypass it. Thank you so much!

EDIT: There are in fact two issues going around. When I typed this into the Sage (version 8.7) on my computer, it came out as written above. However, when I tried this on the webpage version of Sage, there is this 'is_commutative' error as mentioned by tmonteil in the comment, and I don't know how this error comes up either. In any case, it seems to be impossible to compute ideals in this group algebra.

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( 2022-01-30 09:38:19 +0200 )edit

On Sage 9.5.rc0, for the line

sage: I=[T-1]*R;


i got an error:

AttributeError: 'GroupAlgebra_class_with_category' object has no attribute 'is_commutative'


Is your example self-contained ? If yes, which version of Sage are you running ?

( 2022-01-30 10:47:16 +0200 )edit

@tmonteil Thanks for the comment! The version of my Sage is 8.7. I actually also discovered this 'is_commutative' error when I tried it on the online version of Sage. I have edited my question to include this. Thanks!

( 2022-01-30 12:28:10 +0200 )edit