Timeline for Algebra with elements x, y such that r(x)=r(y) for all finite-dimensional modules r
Current License: CC BY-SA 3.0
5 events
| when toggle format | what | by | license | comment | |
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| Sep 11, 2011 at 3:39 | comment | added | Benjamin Steinberg | By the way my answer is a special case of the following fact: if M is a finitely generated monoid which is not residually finite and x,y in M are elements of M not separable in a finite image, then the corresponding elements of the monoid algebra are not separable by finite dimensional reps. This follows from Malcev's theorem that finitely generated linear monoids are residually finite. | |
| Sep 11, 2011 at 0:14 | comment | added | Mariano Suárez-Álvarez | Ah. Sure. What I had in mind is this: Bergman's diamond lemma immediately (there are no ambiguitities at all!) implies that your algebra has a basis of monomials and, in particular, that x and y are different and that yx is not one. | |
| Sep 10, 2011 at 23:53 | comment | added | Benjamin Steinberg | Yes, but it is not immediate from a presentation that the relation xy=1 doesn't imply yx=1. One needs to give an explicit representation showing that xy=1 is possible with yx not equal to 1. | |
| Sep 10, 2011 at 22:14 | comment | added | Mariano Suárez-Álvarez | There is always a faithful representation: the regular one :) | |
| Sep 10, 2011 at 19:26 | history | answered | Benjamin Steinberg | CC BY-SA 3.0 |