Tags » Hopf Algebras

The Math of Shuffling Cards

A magic trick based on the “Perfect Shuffle”. Featuring Professor Federico Ardila. I watched his videos on Hopf Algebras while learning the background material for my honours project on… 65 more words


From the Poincaré group to Heisenberg doubles

There’s a nice geometric way to understand the Heisenberg double of a Hopf algebra, using what one might call its “defining representation(s).” In fact, it’s based on the nice geometric way to understand any semidirect product of groups, so I’ll start with that. 997 more words


A Survey of Extensions of Harmonic Analysis

The standard approach to harmonic analysis is to begin with a  locally compact abelian group (always assumed Hausdorff) and consider its dual (or character) group consisting of all continuous group homomorphisms  One then defines the  961 more words

The Picard scheme of an abelian variety

Let be an algebraically closed field, and a projective variety over . In the previous two posts, we’ve defined the Picard scheme , stated (without proof) the theorem of Grothendieck giving conditions under which it exists, and discussed the infinitesimal structure of (or equivalently of the connected component at the origin). 1,488 more words

Algebraic Geometry

A toy analog of the Sullivan conjecture

In this post, I’d like to describe a toy analog of the Sullivan conjecture. Recall that the Sullivan conjecture considers (pointed) maps from into a finite complex, and states that the space of such is contractible if is finite. 1,069 more words


Oriented cobordism II: comodule structure theorems

The next goal of this series of posts (started here) is to analyze the oriented cobordism spectrum at the prime 2; the main result is that there is a splitting of into a direct sum of copies of (the torsion-free part) and (the torsion-part). 1,425 more words


Formal Lie theory in characteristic zero

Let be a field of characteristic zero. The intuition is that in this case, a Lie algebra is the same data as a “germ” of a Lie group, or of an algebraic group. 1,619 more words