Given a finite dimensional selfinjective quiver algebra A over a finite field (or more generally an arbitrary field). Whats the best way to check if the algebra A has a Hopf algebra structure or not? If we assume the field to be finite it is a finite problem, so there might be some good algorithm? Note that there is a topic with the same question for not necessary finite dimensional algebra. But quiver algebras are much more special and maybe one can find stronger statements. For example the quiver algebra with 1 loop x and relations x^2=0 is a Hopf algebra iff the field has characteristic two. Is there a classification for small dimensions(maybe up to 100) of local finite dimensional (quiver) Hopf algebras?
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2$\begingroup$ The algebra has to be symmetric, and that cuts downs enormously the list of candidates. I doubt there is a classification. $\endgroup$– Mariano Suárez-ÁlvarezCommented Nov 19, 2015 at 20:05
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$\begingroup$ can you give a source that a local quiver algebra that is a hopf algebra, has to be symmetric? its just clear to me that it has to be weakly symmetric. $\endgroup$– MareCommented Nov 19, 2015 at 20:43
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2$\begingroup$ I actually meant Frobenius, not symmetric; this follows frorm a theorem of Larson and Sweedler. $\endgroup$– Mariano Suárez-ÁlvarezCommented Nov 19, 2015 at 20:52
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2$\begingroup$ ok, this i know. note that i wrote "selfinjective" quiver algebra. this is equivalent to frobenius in case of quiver algebras. $\endgroup$– MareCommented Nov 19, 2015 at 20:53
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3$\begingroup$ The paper "Basic Hopf algebras and quantum groups" by Green and Solberg might be helpful. They give some necessary criterias on the underlying quiver in order for the algebra to have a Hopf algebra structure, and also construct some examples of basic Hopf algebras $\endgroup$– SondreCommented Nov 19, 2015 at 22:14
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The paper "algebres de chemin quantiques" by Cibils and Rosso answers exactely that question. Adv in Math 125(2) 1997
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$\begingroup$ Since it is in french, can you state the important result here? $\endgroup$– MareCommented Jun 16, 2019 at 21:35
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$\begingroup$ They given the answer un terms of the quiver. The abstract is also in English. There is also a second paper by the same authors, something like "Hopf quivers", in english. $\endgroup$ Commented Jun 16, 2019 at 21:51
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$\begingroup$ In Journal of Algebra 254 (2002) $\endgroup$ Commented Jun 16, 2019 at 21:53