Is it possible to classify Hopf algebras $H$, over a field $k$, which admit a unique (up to isomorphism) irreducible comodule, namely the trivial $1$-dim comodule $$ k \to k \otimes H, ~~ k \mapsto k \otimes 1_H. $$
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1$\begingroup$ This seems like a very broad question. What sort of ingredients might go into the classification? $\endgroup$– LSpiceCommented Aug 11, 2021 at 21:52
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3$\begingroup$ If you impose the extra condition that $H$ is commutative, then you are asking about the classification of unipotent algebraic groups. As my advisor once told me, "die gibt's wie Sand am Meer". $\endgroup$– Geordie WilliamsonCommented Aug 11, 2021 at 22:04
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2 Answers
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The HAs you are describing are again the connected (=irreducible) ones. I am using the terminology here as in my answer to your previous question: Name for a Hopf algebra whose only grouplike element is the identity?.
Their classification, is in general an open project:
- In the cocommutative case, over a field of $chark=0$, they are all universal enveloping algebras of lie algebras. If $k=\mathbb{C}$, this is by an old result of Milnor and Moore. You can find the statement for any field of char zero at Montgomery's book.
- In the finite dimensional case they appear only over fields of positive characteristic. I think this is an old result of Masuoka (but i do not have the exact reference right now). See arXiv:1309.0286 [math.RA], arXiv:1310.7073 [math.RA] (and the references therein) for recent results on their classification.
answered Aug 12, 2021 at 1:07
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$\begingroup$ Thanks for your answer. Just so I understand precisely - irreducible is a special of connected? Precisely an irreducible Hopf algebras is one that has a single comodule, namely the trivial comodule, and hence it is connected since connected requires the weaker condition of the existence of a single 1-dim comodule. Is this correct. $\endgroup$ Commented Aug 12, 2021 at 18:12
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$\begingroup$ @Spyros, a connected coalgebra has a unique simple comodule, which must be 1-dim. An irreducible one has a unique simple comodule. So at the level of coalgebras, connected are irreducibles (but not necessarily the other way around). So the irreducibles are a wider class. $\endgroup$ Commented Aug 12, 2021 at 20:57
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$\begingroup$ If we are speaking about HAs then the presence of $1$ makes the connected and the irreducibles the same thing. Since then the trivial (1-dim) comodule always makes sense. $\endgroup$ Commented Aug 12, 2021 at 21:02
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There have been several papers published on such Hopf algebras (which are referred to as "connected" Hopf algebras) over the past decade. In particular, over an algebraically closed field of characteristic 0, the discovery of non-commutative + non-cocommutative examples has helped spur on the interest in classifying these algebras. See, for example, the following:
Properties of pointed and connected Hopf algebras of finite Gelfand-Kirillov dimension
Quantum homogeneous spaces of connected Hopf algebras
Connected (graded) Hopf algebras
In general there is no known concrete classification of such connected Hopf algebras. In particular, the last of the papers listed above presents a non-cocommutative, non-commutative connected Hopf algebra which is not isomorphic as an algebra to the universal enveloping algebra of any lie algebra.
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1$\begingroup$ Paul, this is nice. +1. Btw, your third paper is a very interesting one! $\endgroup$ Commented Aug 15, 2021 at 18:10
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$\begingroup$ @KonstantinosKanakoglou thank you! $\endgroup$ Commented Aug 16, 2021 at 18:52