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Jan
26
comment Category of $\textbf{Ring}$.
it's sometimes easier to have a convention that excludes slightly trivial or slightly pathological examples. Dummit and Foote just take a ring to mean something with a unit, they never deal with unit-less rings, and it's extra baggage to carry that around when you aren't using it.
Jan
26
comment Univariate and Matrix Representation of Affine Transformation
the map $\phi$ is implicit in what I'm doing, it's essentially a change-of-basis matrix. if you let $e_i$ be the canonical basis (a column with all 0s except on the i-th row, that's a 1) for $\mathbb{F}^n$, then $\phi : \mathbb{F}^n \to \mathbb{E}^n$ by $e_i \mapsto \{\text{some expression in the t basis I gave}\}$ then it'll do the trick.
Jan
25
comment Univariate and Matrix Representation of Affine Transformation
I'm not 100% sure I'm correct on the last step, but hopefully you see the idea now? :)
Jan
25
comment Application of Category Theory
you might look at these questions: math.stackexchange.com/questions/312605/… and mathoverflow.net/questions/19325/… (heavier maths there)
Jan
14
comment most efficient way to find distinct complimenting subspaces over a finite field
is there an algorithmically efficient way to complete $\{v_{n-k+1},\ldots, v_n\}$ to a basis $\{v_1,\ldots,v_n\}$. Your description solves the problem wonderfully (though I no longer need such a solution!), but I might be too silly to see an way to complete the basis.
Jan
14
comment most efficient way to find distinct complimenting subspaces over a finite field
you're correct, my bad. changed question.
Dec
31
comment Is there a discrete version of non-commutative geometry (yet)?
you might be interested in a paper from Majid arxiv.org/abs/1011.5898 Noncommutative riemannian geometry on graphs. This is not NCG in the manner of connes, though.
Dec
9
comment finding fixed points of frobenius endomorphism of a ring of char > $p$ (not a domain)
this is a very good answer. but it leaves me with a lot of nagging questions: why does the number of cycles = the number of distinct irreducible factors of $x^{p^n} -x$? Why does the number of cycles of length $d$ = the number of such factors of degree $d$? I suppose these things deserve separate questions and I should accept your answer - is that the proper etiquette here?
Jan
5
comment Prove well-definedness of comultiplication and counit of $GL_q(2)$ and $SL_q(2)$.
certainly! google my name on here and you should find my contact info.
Dec
28
comment Prove well-definedness of comultiplication and counit of $GL_q(2)$ and $SL_q(2)$.
I'm working through majid's "primer on quantum groups" (disclaimer: majid is my phd supervisor) along with kassel, and I have a few other texts handy, but they're harder for an undegrad to crack. feel free to email me about it, though.
Dec
17
comment Prove well-definedness of comultiplication and counit of $GL_q(2)$ and $SL_q(2)$.
a word of warning about kassel - it's a great book, but he tends to be a bit "over axiomatic" and it's a bit hard to see the forest through the trees sometimes. For instance, it's a lot easier to just compute $\Delta(\text{det_q})$ then to follow his chain of propositions. It helps to have a few other texts handy to cross-reference these things.
Dec
17
comment Prove well-definedness of comultiplication and counit of $GL_q(2)$ and $SL_q(2)$.
I re-wrote it a bit more coherently, also describing coideals a tad bit more and explaining how he gets the computation. You should ensure you have a good idea of ideals and quotient rings if you don't have that already. coideals were described in ch3. I'm a 2nd year phd student and I find this hopelessly confusing myself. regards!
Dec
17
comment Prove well-definedness of comultiplication and counit of $GL_q(2)$ and $SL_q(2)$.
do you understand how the bialgebra structure on $M_q(2)$ works?
Dec
17
comment proving that a action of hopf algebra k(G) on A implies a G-grading on A
I added some text for clarification, does that help?
Aug
6
comment What are some examples of vector spaces that aren't graded?
well that certainly clears things up. thanks!
Jul
5
comment what simple extensions can be naturally embedded into Conway's algebraic closure of $\mathbb{F}_2$?
this is indeed a helpful answer, and sheds much light on the proof in conway's book I didn't understand (he uses the same technique with that polynomial). In fact it solves my problem for all finite numbers where its possible to solve it. But I'd still like a description of the other subfields, though. I'm going to leave the question open, but if you're interested I'll post a note or another question on the bits I [do not] understand from Conway's book. Thanks! :D
Jul
5
comment About PhD in non-commutative topology
You may want to try reading something like Kalkhali's Very Basic Noncommutative Geometry arxiv.org/abs/math/0408416. Though in my (very limited) experience it's more effective to approach research by trying to tackle a specific problem and learning the tools you need as you go along. In lieu of that, read as much operator algebra and geometry as you can.
Jul
5
comment what simple extensions can be naturally embedded into Conway's algebraic closure of $\mathbb{F}_2$?
Oops. Never mind, such fields are covered by infinite ordinals. This is why I should wait to respond!
Jul
5
comment what simple extensions can be naturally embedded into Conway's algebraic closure of $\mathbb{F}_2$?
One thing is bothering me, though, and this may be a separate question: if $\omega^{\omega^\omega}$ is an algebraic closure, and if an algebraic closure contains all $\mathbb{F}_{2^k}$, even k not a power of 2, then shouldn't $\omega^{\omega^\omega}$ contain a subfield isomorphic to $\mathbb{F}_{2^k}$? Indeed this is hard to reconcile with the multiplication table at me neverendingbooks for say, k=3. Ordinals $2^{2^k}$ only give us a quadratic closure, no?
Jul
5
comment what simple extensions can be naturally embedded into Conway's algebraic closure of $\mathbb{F}_2$?
This is nevertheless a helpful answer! I'll comment more ( or accept) later today when I can read over it a bit more carefully.