Labeled Trees and Localized Automorphisms of the Cuntz Algebras

We initiate a detailed and systematic study of automorphisms of the Cuntz algebras \O_n which preserve both the diagonal and the core $UHF$-subalgebra. A general criterion of invertibility of endomorphisms yielding such automorphisms is given. Combinatorial investigations of endomorphisms related to permutation matrices are presented. Key objects entering this analysis are labeled rooted trees equipped with additional data. Our analysis provides insight into the structure of {\rm Aut}(\O_n) and leads to numerous new examples. In particular, we completely classify all such automorphisms of ${\mathcal O}_2$ for the permutation unitaries in $\otimes^4 M_2$. We show that the subgroup of {\rm Out}(\O_2) generated by these automorphisms contains a copy of the infinite dihedral group ${\mathbb Z} \rtimes {\mathbb Z}_2$.Comment: 35 pages, slight changes, to appear on Trans. Amer. Math. So

Counting Semilinear Endomorphisms Over Finite Fields

For a finite field k and a triple of integers g \ge r \ge s \ge 0, we count the number of semilinear endomorphisms of a g-dimensional k-vector space which have rank r and stable rank s. Such endomorphisms show up naturally in the classification of finite flat group schemes of p-power order over k which are killed by p and have p-rank s, via Dieudonne theory

Quantum Hamiltonian reduction of W-algebras and category O

W-algebras are a class of non-commutative algebras related to the classical universal enveloping algebras. They can be defined as a subquotient of U(g) related to a choice of nilpotent element e and compatible nilpotent subalgebra m. The definition is a quantum analogue of the classical construction of Hamiltonian reduction. We define a quantum version of Hamiltonian reduction by stages and use it to construct intermediate reductions between different W-algebras U(g,e) in type A.This allows us to express the W-algebra U(g,e') as a subquotient of U(g,e) for nilpotent elements e' covering e. It also produces a collection of (U(g,e),U(g,e'))-bimodules analogous to the generalised Gel'fand-Graev modules used in the classical definition of the W-algebra; these can be used to obtain adjoint functors between the corresponding module categories. The category of modules over a W-algebra has a full subcategory defined in a parallel fashion to that of the Bernstein-Gel'fand-Gel'fand (BGG) category O; this version of category O(e) for W-algebras is equivalent to an infinitesimal block of O by an argument of Mili\v{c}i\'{c} and Soergel. We therefore construct analogues of the translation functors between the different blocks of O, in this case being functors between the categories O(e) for different W-algebras U(g,e). This follows an argument of Losev, and realises the category O(e') as equivalent to a full subcategory of the category O(e) where e' is greater than e in the refinement ordering.Comment: University of Toronto PhD thesis, defended July 2014, 57 page

Isospectral pairs of metrics on balls, spheres, and other manifolds with different local geometries

The first isospectral pairs of metrics are constructed on balls and spheres. This long standing problem, concerning the existence of such pairs, has been solved by a new method called "Anticommutator Technique." Among the wide range of such pairs, the most striking examples are provided on (4k-1)-dimensional spheres, where k > 2. One of these metrics is homogeneous (since it is the metric on the geodesic sphere of a 2-point homogeneous space), while the other is locally inhomogeneous. These examples demonstrate the surprising fact that no information about the isometries is encoded in the spectrum of Laplacian acting on functions. In other words, "The group of isometries, even the local homogeneity property, is lost to the "Non-Audible" in the debate of "Audible versus Non-Audible Geometry"."Comment: 43 pages. After retrieving source, read README file or type tex whole to typeset (in Unix