Hall algebra for non-abelian p-groups ? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-21T19:17:50Z http://mathoverflow.net/feeds/question/107537 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/107537/hall-algebra-for-non-abelian-p-groups Hall algebra for non-abelian p-groups ? Alexander Chervov 2012-09-19T08:12:55Z 2012-09-19T08:12:55Z <p>According to <a href="http://en.wikipedia.org/wiki/Hall_algebra" rel="nofollow">WP article on Hall algebras</a> one counts the number of abelian subgroups in abelian group with fixed type of subgroup, group, quotient.</p> <p>Two things are claimed:</p> <p>1) These numbers are polynomials in "p"</p> <p>2) Using these numbers one naturally defines the algebra structure on the isomorphism classes of abelian groups which appears to be associative and commutative.</p> <p><strong>Question:</strong> What happens if we consider all p-groups, not just abelian one ? Will same/similar claims be true ?</p> <p>PS</p> <p>The natural context for the question seems to me some categories with finite number of exact triples A->B->C for fixed A,B,C. So natural generalization - what the properties of the categories for which we can define associative algebra ? commutative algebra ? </p>