bio | website | iaz.uni-stuttgart.de/… |
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location | Stuttgart | |
age | 28 | |
visits | member for | 3 years, 11 months |
seen | May 21 at 7:54 | |
stats | profile views | 1,630 |
Post-Doc working in representation theory
Apr 7 |
comment |
Why is the A6 preprojective algebra of wild representation type?
There is a proof in [Erdmann-Skowronski: The stable Calabi-Yau dimension of tame symmetric algebras, Lemma 3.5 and Theorem 3.7] |
Apr 2 |
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The Jacobson radical of an infinite dimensional algebra
There is a general statement on the Jacobson radical of the path algebra of a quiver (without relations), see this mathoverflow question. |
Apr 2 |
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Why Jacobson, but not the left (right) maximals individually?
This statement is also proved in a slightly greater generality in [Coelho, Liu: Generalized path algebras. Interactions between ring theory and representations of algebras (Murcia), 53–66, Lecture Notes in Pure and Appl. Math., 210, Dekker, New York, 2000. MR1758401]. |
Mar 25 |
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References for the bicategory of ring-bimodule pairs
@PavelSafronov That's a nice observation. If I understand it correctly from the brief look I took at the definition it should be a lax slice $2$-category over $\mathbb{Z}$ (resp. $k$). |
Mar 25 |
asked | References for the bicategory of ring-bimodule pairs |
Mar 16 |
answered | Kaplansky's 6th conjecture: dim(Irrep) | dim(algebra) - for semi-simple Hopf algebras |
Mar 16 |
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Kaplansky's 6th conjecture: dim(Irrep) | dim(algebra) - for semi-simple Hopf algebras
Probably clear to most readers, but this is false in positive characteristic. A counterexample is $\operatorname{SL}(2,p)$ with $p$ odd as explained in Curtis-Reiner, (17.17). |
Mar 14 |
awarded | Popular Question |
Feb 20 |
awarded | Electorate |
Oct 21 |
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Higher vector spaces
@ChrisSchommer-Pries Thanks for pointing out this mistake. Of course the category should be cocomplete, otherwise there is the question what compact should mean if you don't have coproducts. I don't know of a theorem describing what properties describe finite dimensional vector spaces (or more general finitely generated, finitely presented, or coherent modules over a ring). |
Oct 21 |
revised |
Higher vector spaces
added the assumption of being cocomplete |
Oct 18 |
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Higher vector spaces
@Najib Idrissi Yes, I changed accordingly. |
Oct 18 |
revised |
Higher vector spaces
added 18 characters in body |
Oct 16 |
answered | Higher vector spaces |
Oct 16 |
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Higher vector spaces
Yes, sure. I was mixing up the tensor products, sorry. |
Oct 16 |
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Higher vector spaces
One needs commutative or an $(A,B^{op})$-bimodule. Otherwise the tensor functor will go to left modules instead of right modules. The image should be described by using a $k$-linear version of Morita's theorem: There should exist a compact progenitor. What do you mean by fully faithful for a $2$-functor? |
Jul 3 |
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Is there a homological way to compute quiver presentations?
Oh, sorry again: It is the $m$-th tensor power on the right hand side. This you should take to have arbitrary large path lengths in your relations. |
Jul 3 |
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Is there a homological way to compute quiver presentations?
Oh, the two $m$'s have a different meaning, unfortunately. The $m$ on the right hand side is just the summation index. The $m$ on the left hand side is the $A_\infty$-multiplication. A reference is [Keller: A-infinity algebras in representation theory], in the slightly different context of graded algebras [Lu, Palmieri, Wu, Zhang: A-infinity structure on Ext-algebras], or specifically about (a generalisation of) acyclic quiver, [Koenig, Külshammer, Ovsienko: Quasi-hereditary algebras, exact Borel subalgebras, A-infinity categories and boxes], taking standards to be simples, and ignoring box |
Jul 2 |
awarded | Curious |
Jul 2 |
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Is there a homological way to compute quiver presentations?
I think Dag Madsen is suggesting the homological approach of computing the $A_\infty$-structure of the $\operatorname{Ext}$-algebra of the simples. This gives the quiver and relations. The relations are given by the image of a map $Dm:D\operatorname{Ext}^2(S,S)\to \bigoplus (D\operatorname{Ext}^1(S,S))^m$, where $S$ is the direct sum of all the simples. |