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Hi,

I would like to know what pointed Hopf algebras are and why it is that they are important.

Thank you.

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While I don't deem the question "what is a pointed Hopf algebra" appropriate, I sympathize with the second one. Back when I was attending a Hopf algebra course, this was exactly my question, and I didn't obtain a good (for me!) answer to it until I studied combinatorial Hopf algebras.

Many Hopf algebras that appear in combinatorics (the tensor and shuffle Hopf algebras of a vector space, as well as the Hopf algebras $\mathbf{Symm}$, $\mathbf{QSymm}$, $\mathbf{NSymm}$, Loday-Ronco, Malvenuto-Reutenauer, trees, ordered trees, ...) are naturally connected graded or at least connected filtered (like the universal enveloping algebra of a Lie algebra). There is a lot to be said about this kind of Hopf algebras. Most importantly, their "connected filtered" property helps proving things about them; it more or less gives us a way to proceed by induction over the degree.

However, at one moment, non-connected graded and filtered Hopf algebras started to appear in combinatorics: e.g., the $\widetilde{\mathcal H}_{\mathcal T}$ in Dominique Manchon's "Hopf algebras, from basics to applications to renormalization". Just knowing that they are graded does not help (we could have the whole Hopf algebra concentrated in degree $0$). So we need some other condition that these Hopf algebras satisfy. One such condition (which is trivially satisfied in the case of $\widetilde{\mathcal H}_{\mathcal T}$) is that the $0$-th part of the grading (or, more generally, of the filtration) is spanned by grouplike elements. In other words, every simple subcoalgebra of the $0$-th part of the filtration is $1$-dimensional. (Unfortunately, this condition is not geometric, i. e., it can change under an algebraic extension of the ground field. But it is a good start, and for the cases that appear in combinatorics, it holds over any field, because the grouplike elements usually belong to the combinatorial basis.)

Now assume you are given just a random Hopf algebra without filtration. Can you use any of these things that you have proven for connected filtered Hopf algebras? Well, you can canonically define a filtration on it (at least over a field!), the so-called coradical filtration. If the coradical $C_0$ (the $0$-th part of the filtration) is $1$-dimensional, then your Hopf algebra becomes connected filtered, and you have won. Hopf algebras like this are said to be irreducible. If the coradical $C_0$ is spanned by grouplike elements, then you are in the second case discussed above, and the Hopf algebra is said to be pointed.

It is easy to see (using Artin-Wedderburn) that cocommutative Hopf algebras over an algebraically closed field are pointed (while clearly not being always irreducible!), so "pointed" (or, more precisely, "pointed after base change to the algebraic closure") is a kind of weakening of "cocommutative", and you can try to generalize everything you know about cocommutative Hopf algebras to the pointed case: e. g., Cartier-Milnor-Moore becomes Cartier-Kostant.

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A Hopf algebra is called pointed if all its simple left (or right) comodules are one-dimensional.The quantized enveloping algebras and Lusztig's small quantum groups are examples of pointed Hopf algebras.

A good survey about finite-dimensional pointed Hopf algebras (and their classification project) is:

Andruskiewitsch, Nicolás; Schneider, Hans-Jürgen. Pointed Hopf algebras. New directions in Hopf algebras, 1--68, Math. Sci. Res. Inst. Publ., 43, Cambridge Univ. Press, Cambridge, 2002. MR1913436 (2003e:16043)

Here you find the paper.

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I would add to Vendramin's answer that for finite-dimensional semisimple pointed Hopf algebras $H$, the representation category $Rep(H)$ is much like a group: all the simple objects (irreps.) are invertible. As such these are essentially classified, and the \emph{non-semisimple} case is of greatest interest. –  Eric Rowell Jan 25 '12 at 15:53
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