Given an algebra, can it be realized as a block of a Hopf algebra? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-22T00:43:48Z http://mathoverflow.net/feeds/question/68274 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/68274/given-an-algebra-can-it-be-realized-as-a-block-of-a-hopf-algebra Given an algebra, can it be realized as a block of a Hopf algebra? Julian Kuelshammer 2011-06-20T10:43:29Z 2011-08-31T21:45:17Z <p>During a classification problem I came across a set of algebras given as the path algebra of a quiver with relations. As an example the local ones: $k\langle x,y\rangle/x^2,y^2, xy-qyx$, where $q\in k$, sometimes called truncated quantum plane. </p> <p>My question is whether there are necessary/sufficient criteria for (these) algebras to be a block of a Hopf algebra (up to Morita equivalence). For example, $q$ obviously has to be a root of unity, because otherwise the Nakayama automorphism will not be of finite order. A sufficient criteria is $q=1$, but I don't know of other instances. (Maybe it also depends on the characteristic of the underlying field).</p> <p>Weaker results would also be of interest to me, for example [Farnsteiner: Polyhedral groups, McKay quivers, and the finite algebraic groups with tame principal blocks, Proposition 7.4.3] shows that they will not be the principal block of a cocommutative Hopf algebra if $char k \geq 3$.</p>