Is there a classifications of (connected+ finite-dimensional) algebras A that are Hopf algebras with indecomposable projective modules $P_1,...,P_n$, such that every algebra of the form $B=End_A( P_1^{m_1} \oplus ... \oplus P_n^{m_n})$ (which is morita equivalent to A) is also a Hopf algebra as long as one $m_i=1$?
Classification of special Hopf algebras
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representation-theory
hopf-algebras
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1The paper Green-Solberg: Basic Hopf algebras and quantum groups gives a result which quivers can occur. If by "quiver algebra" you mean without relation, then there is a necessary and sufficient condition in Cibils-Rosso: Algèbres des chemins quantiques. – 2017-02-14
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0@JulianKuelshammer But quivers without relations are never Hopf algebras expect for the field (assuming finite dimensional algebras). – 2017-02-14
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1The paper by Cibils-Rosso considers also path algebras of infinite quivers. – 2017-02-14
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0By the way, I would be sceptical that there is an algebra $A$ such that every algebra $B$ Morita equivalent to $A$ has a Hopf algebra structure. Have you checked whether this holds for $A=k$ even? – 2017-02-14
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2As a Hopf algebra needs to have a $1$-dimensional module, no finite dimensional algebra can have the property that every Morita equivalent algebra has a Hopf algebra structure. – 2017-02-14
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0Thats a good point, I modified the question so that it hopefully makes more sense. – 2017-02-19