Kaplansky's 6th conjecture: dim(Irrep) | dim(algebra) - for semi-simple Hopf algebras - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-24T10:20:19Z http://mathoverflow.net/feeds/question/108404 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/108404/kaplanskys-6th-conjecture-dimirrep-dimalgebra-for-semi-simple-hopf-alge Kaplansky's 6th conjecture: dim(Irrep) | dim(algebra) - for semi-simple Hopf algebras warren 2012-09-29T12:47:22Z 2012-09-29T16:10:16Z <p>Let \$H\$ be a semisimple Hopf algebra. One of the Kaplansky's conjectures states that the dimension of any irreducible \$H\$-module divides the dimension of \$H\$. </p> <p>In which cases the conjecture is known to be true?</p> http://mathoverflow.net/questions/108404/kaplanskys-6th-conjecture-dimirrep-dimalgebra-for-semi-simple-hopf-alge/108405#108405 Answer by Alexander Chervov for Kaplansky's 6th conjecture: dim(Irrep) | dim(algebra) - for semi-simple Hopf algebras Alexander Chervov 2012-09-29T12:53:48Z 2012-09-29T12:53:48Z <p>From <a href="http://www2.math.technion.ac.il/~gelaki/research_summary.pdf" rel="nofollow">Shlomo Gelaki research statement</a> (which is nice survey, by the way):</p> <blockquote> <p>We also proved that the dimension of an irreducible representation of a semisimple Hopf algebra H, which is either quasitriangular or cotriangular, divides the dimension of H. This result partially answers a celebrated conjecture of Kaplansky, which is still open.</p> </blockquote> http://mathoverflow.net/questions/108404/kaplanskys-6th-conjecture-dimirrep-dimalgebra-for-semi-simple-hopf-alge/108412#108412 Answer by Leandro Vendramin for Kaplansky's 6th conjecture: dim(Irrep) | dim(algebra) - for semi-simple Hopf algebras Leandro Vendramin 2012-09-29T16:10:16Z 2012-09-29T16:10:16Z <p>Yorck Sommerhäuser has a very nice <a href="http://www.mathematik.uni-muenchen.de/~sommerh/Publikationen/KaplConjwww/KaplConjRev.ps" rel="nofollow">survey</a> about Kaplansky's conjectures. Section 6 is devoted to Kaplansky's 6th conjecture. </p> <p>In Sommerhäuser's survey it is mentioned that Richmond and Nichols proved that the conjecture is true if the simple module has dimension two:</p> <blockquote> <p>Theorem (Nichols &amp; Richmond). The dimension of a semisimple Hopf algebra over \$\mathbb{C}\$ is even if the Hopf algebra has a simple module of dimension 2.</p> </blockquote> <p>In Sommerhäuser's survey it is also mentioned that Montgomery and Witherspoon proved that Kaplansky's conjecture holds if it holds for a subalgebra.</p> <p>In this paper </p> <p>Cohen, Miriam; Gelaki, Shlomo; Westreich, Sara. Hopf algebras. Handbook of algebra. Vol. 4, 173--239, Handb. Algebr., 4, Elsevier/North-Holland, Amsterdam, 2006. MR2523421 (2010j:16076), <a href="http://www.sciencedirect.com/science/article/pii/S1570795406800070" rel="nofollow">link</a></p> <p>it is written that Kaplansky's conjecture has been proved </p> <ul> <li><p>if \$H\$ is triangular,</p></li> <li><p>if \$H\$ is semisolvable, </p></li> <li><p>if \$H\$ is cotriangular,</p></li> <li><p>if \$R(H)\$ is central in \$H^*\$, where \$R(H)\$ is the span in \$H^*\$ of all the characters on \$H\$.</p></li> </ul>