Generalized Twisted Quantum Doubles and the McKay Correspondence

We consider a class of quasi-Hopf algebras which we call \emph{generalized twisted quantum doubles}. They are abelian extensions H = \mb{C}[\bar{G}] \bowtie \mb{C}[G] ($G$ is a finite group and $\bar{G}$ a homomorphic image), possibly twisted by a 3-cocycle, and are a natural generalization of the twisted quantum double construction of Dijkgraaf, Pasquier and Roche. We show that if $G$ is a subgroup of SU_2(\mb{C}) then $H$ exhibits an orbifold McKay Correspondence: certain fusion rules of $H$ define a graph with connected components indexed by conjugacy classes of $\bar{G}$, each connected component being an extended affine Diagram of type ADE whose McKay correspondent is the subgroup of $G$ stabilizing an element in the conjugacy class. This reduces to the original McKay Correspondence when $\bar{G} = 1$.Comment: 5 figure

Using Composition Techniques to Improve Classroom Instruction and Students’ Understanding of Proof

This paper describes an effort to incorporate standard composition exercises into a sophomore-level discrete mathematics class. It provides an example of how peer review can be integrated with a mathematical curriculum through the writing of proofs

Is Hollywood Good for Mathematics? (a brief discussion).

Mathematics has played a role in many recent films, such as Good Will Hunting, A Beautiful Mind, and Proof. But how exactly is mathematics being portrayed? What ramifications does this portrayal have on our ”students,” broadly defined? In Good Will Hunting, mathematics is Will’s ticket out of his lower-class life and into a prestigious existence. In A Beautiful Mind, mathematical talent and schizophrenia are identified, at least in part. In Proof, mathematics not only links a daughter to her late father, but it also provides a backdrop for much of the story’s conflict. In this discussion, we will begin by analyzing the role mathematics plays in various films. Then, we will examine possible effects these films have on public (and thus, student) perceptions of mathematics and mathematicians

A family of isomorphic fusion algebras of twisted quantum doubles of finite groups

Let Dω(G) be the twisted quantum double of a finite group, G, where ω∈Z3(G,C∗). For each n∈N, there exists an ω such that D(G) and Dω(E) have isomorphic fusion algebras, where G is an extraspecial 2-group with 22n+1 elements, and E is an elementary abelian group with |E|=|G|

An explicit fusion algebra isomorphism for twisted quantum doubles of finite groups

We exhibit an isomorphism between the fusion algebra of the quantum double of an extraspecial p-group, where p is an odd prime, and the fusion algebra of a twisted quantum double of an elementary abelian group of the same order

Fusion rules for abelian extensions of Hopf algebras

We investigate the representation theory and fusion rules of a class of cocentral abelian (quasi-)Hopf extensions of Hopf algebras which includes twisted (generalized) quantum doubles of finite groups, and a certain quasi-Hopf algebra of Schauenburg associated to group-theoretical fusion categories. We then present a nontrivial example with noncommutative fusion rules

Does Spelling Count? Reflections on Writing in the Mathematics Classroom

The written word is scarce in the stereotypical mathematics course, which instead emphasizes the routine solving of problems. But having students explain their solutions in writing can pave the way to more critical mathematical thinking. This talk will examine the use of writing as a tool to learn mathematics. We will reflect on: reasons to incorporate writing in a mathematics class; which courses lend themselves to writing assignments; what types of assignments to use; samples of student work; and whether using writing can actually improve student learning. Depending on the assignments, writing can also incorporate several “Basic Principles” of AMATYC’s Beyond Crossroads, such as Innovation, Inquiry, Relevance, and Assessment