Timeline for Hopf dual of the Hopf dual
Current License: CC BY-SA 4.0
13 events
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| Aug 31, 2020 at 19:50 | history | edited | darij grinberg | CC BY-SA 4.0 |
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| May 9, 2019 at 23:25 | comment | added | Marco Farinati | If G is a simple group of cardial grater than the cardinal of k, then k[G] it cannot have a finite dimensional represetation, just by cardinality reasons. | |
| Aug 25, 2018 at 17:00 | history | bounty ended | CommunityBot | ||
| Aug 23, 2018 at 15:51 | vote | accept | Nadia SUSY | ||
| Aug 19, 2018 at 14:49 | comment | added | Ben Wieland | There is an easy example of a group with no finite dimensional representations: $A_\infty=\bigcup A_n=$ the set of (even) permutations of $\mathbb Z$ with finite support. A nontrivial representation must be nontrivial when restricted to sufficiently large $A_n$, but the necessary dimension grows with $n$. This group is not finitely generated, but if you throw in the shift, you get $\mathbb Z\ltimes A_\infty$, which is finitely generated and has all finite dimensional representations factor through $\mathbb Z$. Finite presentation is harder, the Higman group. See also Baumslag-Solitar | |
| Aug 18, 2018 at 21:00 | comment | added | darij grinberg | @SamHopkins: Not sure about that... | |
| Aug 18, 2018 at 20:52 | comment | added | Sam Hopkins | So are there any infinite-dimensional algebras with a positive answer? | |
| Aug 18, 2018 at 20:43 | history | edited | darij grinberg | CC BY-SA 4.0 |
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| Aug 18, 2018 at 10:31 | comment | added | darij grinberg | Ah, yes: Lemma 2.7 in Robert J. Blattner, Miriam Cohen and Susan Montgomery, Crossed products and inner actions of Hopf algebras, Trans. Amer. Math. Soc. 298 (1986), pp. 671--711. | |
| Aug 18, 2018 at 4:21 | comment | added | zibadawa timmy | There's a 1986 paper of Blattner, Cohen, and Montgomery that shows that if $K$ is an infinite field of cardinality greater than $k$ and $G=PSL_2(K)$, then the Hopf dual of $kG$ is one-dimensional. Montgomery's book even points out looking at an exercise in a 1980 (1977 for the original in Japanese) book by Abe for this. | |
| Aug 17, 2018 at 20:33 | history | edited | darij grinberg | CC BY-SA 4.0 |
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| Aug 17, 2018 at 17:02 | history | edited | darij grinberg | CC BY-SA 4.0 |
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| Aug 17, 2018 at 16:11 | history | answered | darij grinberg | CC BY-SA 4.0 |