Timeline for Name for a Hopf algebra admitting no non-trivial 1-dimensional comodule
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| Jun 4, 2021 at 22:31 | history | edited | Konstantinos Kanakoglou | CC BY-SA 4.0 |
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| Jun 2, 2021 at 20:37 | comment | added | user164898 | I do not know a name for them, but I spent a while thinking through Dmitry Vaintrob's answer to this question, and while I did not attempt to write out a careful proof, I came away thinking that Dmitry's answer appears correct, at least for finite-dim'l Hopf algebras (and for finite-type graded Hopf algebras). Dualizing Dmitry's answer, I would expect that "algebra with nilpotent abelianization" is the name you're asking for. | |
| Jun 2, 2021 at 20:07 | comment | added | Konstantinos Kanakoglou | @A.S., motivated by your comment i was thinking .. do you know if there is a name for such algebras? I mean algebras satisfying the dual property to that described in the OP; that is algebras admitting no non-trivial 1-dim modules (but possibly admitting higher dim simple modules)? | |
| May 29, 2021 at 12:38 | comment | added | Konstantinos Kanakoglou | @A.S. thanks for mentioning an example. yes this is what i meant: there are indeed other HAs (non-connected) which have the desired property. But in any case the connected ones still form a concrete class of HAs which has the required property (though in a more special form). | |
| May 29, 2021 at 4:34 | comment | added | user164898 | You are right that connectedness is too restricted. Let $C$ denote the complex numbers, and consider the Hopf algebra $C[A_5]^*$, the $C$-linear dual of the group algebra of the alternating group $A_5$. One-dimensional $C[A_5]^*$-comodules are given by one-dimensional complex representations of $A_5$, and from the character table (or from simplicity of $A_5$) you know that there is only one, up to isomorphism. So $C[A_5]^*$ is of the kind that the original question asks for. But $C[A_5]^*$ is not a connected Hopf algebra, since any grading on it must put the entire Hopf algebra in degree zero. | |
| May 28, 2021 at 23:04 | history | edited | Konstantinos Kanakoglou | CC BY-SA 4.0 |
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| May 28, 2021 at 22:52 | history | edited | Konstantinos Kanakoglou | CC BY-SA 4.0 |
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| May 28, 2021 at 22:43 | history | edited | Konstantinos Kanakoglou | CC BY-SA 4.0 |
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| May 28, 2021 at 22:15 | history | edited | Konstantinos Kanakoglou | CC BY-SA 4.0 |
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| May 28, 2021 at 21:32 | history | edited | Konstantinos Kanakoglou | CC BY-SA 4.0 |
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| May 28, 2021 at 21:15 | history | edited | Konstantinos Kanakoglou | CC BY-SA 4.0 |
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| May 28, 2021 at 20:54 | history | answered | Konstantinos Kanakoglou | CC BY-SA 4.0 |