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Is there a classification of indecomposable non-semisimple finite dimensional Hopf algebras with exactly two 1 dimensional modules? If not, is there one when all simple modules are 1 dimensional and there are only two simple modules? If there is no full answer, I'm also interested in some examples of such Hopf algebras. By the way: is there a formula for the number of simple modules involving the structure of the group of grouplike elements like in the case of a group algebra? Thank you for answers. edit: Im also interested for non-semisimple indecomposable k-algebras whose basic algebra is a hopf algebra(in general or more special with only 2 simple modules).maybe there is some kind of criteria?

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Over $\mathbb{C}$? –  Simon Wadsley Mar 19 '13 at 21:21
    
no im interested in any field.but if you have some examples over C you can post them too of course –  trew Mar 19 '13 at 21:30
    
Concerning the basic Hopf algebras question, there is a paper by Green and Solberg: Basic Hopf algebras and quantum groups. –  Julian Kuelshammer Mar 20 '13 at 8:38
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Not really engaging with your question, but a family of examples over a field $k$ of characteristic $p$ (for $p$ odd) is the group algebra $kG$ for $G$ any group of order $2p^n$. The Sylow $p$-subgroup of $G$ must be normal as it has index $2$ and must act trivially on any simple module. Thus the simple modules factor through $kC_2$ which obviously has two (1-dimensional) simple modules. –  Simon Wadsley Mar 20 '13 at 11:12
    
As Simon remarks, there are numerous examples involving group algebras of finite groups. And over the complex field the symmetric groups provide natural examples. At any rate, the answer to the question in your first line seems to be no: the Hopf algebras of finite dimension aren't classified. –  Jim Humphreys Mar 20 '13 at 20:49

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