Wikipedia:Reference desk/Archives/Mathematics/2010 September 16

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September 16

Semisimple modules completely determined by character trace map?

Hello,

I have been learning quite some representation theory, but I have stumbled upon some problems, since most authors state the results for group algebras over the complex numbers, while I am interested in semisimple algebras and modules in general.

Therefore:

let A be any algebra over any field F, and let M_1,M_2 be semisimple modules over A. Is it true that M_1 and M_2 are isomorphic if and only if the two trace maps from A to F are equal?

I know this is true if $F$ is the field of complex numbers, but I am interested in the most general case.

Many thanks, Evilbu (talk)

How about this? Let A=F=the field of order two, M₁=A⊕A (trace always 0) and M₂=A⊕A⊕A⊕A (trace also always 0), but M₁ and M₃ are not isomorphic. The reason for starting with modules over the complex numbers is the theory is much simpler in that case and you at least get Burnside's theorem. There is an entire field of modular representation theory which is more general.--RDBury (talk) 16:29, 16 September 2010 (UTC)

Thanks, that is indeed a quite simple counterexample? Are there any counterexamples where the two modules have the same dimension over chosen field? Evilbu (talk)

Rado numbers

Consider the integer linear equation $\sum _{i=1}^{n}c_{i}x_{i}=0$ where $c_{i}(\neq 0)\in Z$. Supposing it is given that there is a natural number N such that, if {1,2...N} is partitioned in two sets, one of these always contains a solution of the equation. The minimal such N is called the Rado number of the equation. I am looking for general bounds on such N, in the cases where it exists. Where can I possibly find such results. Thanks-Shahab (talk) 11:32, 16 September 2010 (UTC)

Surface Normal

Given that C is closed cure that is the boundary of the triangle T with vertices (1,0,0), (0,1,0) and (0,0,1), I have to directly evaluate, without using Stokes' Theorem, $\iint _{T}\nabla \times \mathbf {A} \cdot d\mathbf {S}$, with $\mathbf {A} =(z^{2}-y^{2},x^{2}-z^{2},y^{2}-z^{2})$, by showing that the normal can be written in the form (1,1,1)dy dz but I don't know how to find the normal. I think I'm just being stupid and should know it but I just can't think of it. Can someone help please? Thanks. asyndeton talk 16:29, 16 September 2010 (UTC)

The equation of the plane through the three given points is x+y+z=1, so the normal is ∇(x+y+z).--RDBury (talk) 16:33, 16 September 2010 (UTC)

So simple. Cheers. asyndeton talk 16:36, 16 September 2010 (UTC)