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Feb
4
comment Can the ribbon category of f.d. reps of $\mathcal{U}_q(\mathfrak{sl}(2))$ be modified so the twist is trivial on the vector representation?
The point here is roughly that central character gives a grading on representations, and so elements of the center of the corresponding Lie group give you the ways of changing ribbon structure. For SL the center is finite and so you only get a few ribbon structures, but for GL the center is the scalars and so you can change ribbon structure at the vector rep however you want. The ribbon structure won't be trivial everywhere, in particular the quantum determinant will no longer have trivial ribbon element (which is why this ribbon structure doesn't descend to SL).
Jan
29
accepted Technical issue in the approach to Lie groups taken in Brian C. Hall's book
Jan
29
awarded  gr.group-theory
Jan
28
revised Technical issue in the approach to Lie groups taken in Brian C. Hall's book
added 11 characters in body
Jan
28
answered Technical issue in the approach to Lie groups taken in Brian C. Hall's book
Jan
28
comment Technical issue in the approach to Lie groups taken in Brian C. Hall's book
@TheoJohnson-Freyd: Thanks! It was just a bit weird that I kept finding that same detail skipped over and over again when trying to look this up.
Jan
27
answered Technical issue in the approach to Lie groups taken in Brian C. Hall's book
Jan
27
awarded  Nice Question
Jan
27
awarded  Inquisitive
Jan
27
comment Technical issue in the approach to Lie groups taken in Brian C. Hall's book
@VítTuček: I think you're right, that works. The key fact is that the adjoint form is automatically a closed subgroup of $\mathrm{GL}(\mathfrak{g})$ when $\mathfrak{g}$ is simple.
Jan
27
comment Technical issue in the approach to Lie groups taken in Brian C. Hall's book
@TheoJohnson-Freyd: Of course you're right that $\mathrm{SL}_2(R)$ is not compact, but it shows that you need to use compactness somehow in that step. While you're here, why is the image of Ad a closed subgroup in Prop 7.2.1.6 from the Lie groups notes on your webpage?
Jan
26
comment Technical issue in the approach to Lie groups taken in Brian C. Hall's book
@VítTuček: Does that characterization always yield a matrix compact Lie group?
Jan
26
comment Technical issue in the approach to Lie groups taken in Brian C. Hall's book
One approach would be to show that compact matrix groups do have matrix universal covers, but as far as I can tell that's hard (Peter-Weyl). Another possible approach is to show that the root system breaks up as a direct sum and then use the summand root systems to explicitly realize each of the summands as a compact-semisimple algebra.
Jan
26
revised Technical issue in the approach to Lie groups taken in Brian C. Hall's book
added 2 characters in body
Jan
26
asked Technical issue in the approach to Lie groups taken in Brian C. Hall's book
Jan
8
comment Why are quantum groups so called?
@EdwardHughes: I really like the explanation given in the first few chapters of Kassel's book which explains how quantum 2-by-2 matrices act on the quantum plane in a way that naturally deforms the action of matrices on the ordinary plane. It's not really short enough for MO because everything needs to be rewritten as Hopf algebras coacting on algebras of functions. But Kassel's book is pretty accessible.
Jan
8
comment Why are quantum groups so called?
I don't think there's any benefit to being excessively rigid about what is or is not a quantum group. The question was about the motivation and meaning of the term, not about all situations people use it in.
Jan
8
awarded  Nice Answer
Jan
7
answered Why are quantum groups so called?
Dec
27
comment Restriction of $H^3(M_{24}, U(1))$ to $M_{12} \rtimes \mathbb{Z}_2$
@NoamD.Elkies: The examples I care about are a lot smaller (small finite quotients of PSL_2(Z) restricting to the usual Z/2 and Z/3 subgroups).