I would like to know some examples of connected Hopf algebras which has infinite Gelfand-Kirillov dimension but with primitive space finite dimensional. Any commments are welcome!
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I do not know a concrete example but a (somewhat more refined) version of your question is listed as an open problem (see: question 4.5.3, p. 59) in the following Phd thesis: http://dc.uwm.edu/cgi/viewcontent.cgi?article=2733&context=etd – Konstantinos Kanakoglou Aug 29 '21 at 20:29
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@KonstantinosKanakoglou Thanks a lot! – G.-S. Zhou Aug 30 '21 at 00:25
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I suggest you ping user @Paul Gilmartin on this question. He has various interesting works on connected HAs and maybe (?) he has something more to say on this. – Konstantinos Kanakoglou Aug 31 '21 at 18:07