arXiv subjects

David Jordan

Publications and source records attributed to David Jordan.

41 records · Page 3Linked to original sources

arXiv2009

New results on the lower central series quotients of a free associative algebra

We continue the study of the lower central series and its associated graded components for a free associative algebra with n generators, as initiated by B. Feigin and B. Shoikhet. We establish a linear bound on the degree of tensor field modules appearing in the Jordan-Hoelder series of each graded component, which is conjecturally tight. We also bound the leading coefficient of the Hilbert polynomial of each graded component. As applications, we confirm conjectures of P. Etingof and B. Shoikhet concerning the structure of the third graded component.

math.RA↗

arXiv2009

On the classification of certain fusion categories

We advance the classification of fusion categories in two directions. Firstly, we completely classify integral fusion categories -- and consequently, semi-simple Hopf algebras -- of dimension $pq^2$, where $p$ and $q$ are distinct primes. This case is especially interesting because it is the simplest class of dimensions where not all integral fusion categories are group-theoretical. Secondly, we classify a certain family of $\ZZ/3\ZZ$-graded fusion categories, which are generalizations of the $\ZZ/2\ZZ$-graded Tambara-Yamagami categories. Our proofs are based on the recently developed theory of extensions of fusion categories.

math.QA↗

arXiv2008

Quantum D-modules, elliptic braid groups, and double affine Hecke algebras

We build representations of the elliptic braid group from the data of a quantum D-module M over a ribbon Hopf algebra U. The construction is modelled on, and generalizes, similar constructions by Lyubashenko and Majid, and also certain geometric constructions of Calaque, Enriquez, and Etingof concerning trigonometric Cherednik algebras. In this context, the former construction is the special case where M is the basic representation, while the latter construction can be recovered as a quasi-classical limit of U=U_t(sl_N), as t limits 1. In the latter case, we produce representations of the double affine Hecke algebra of type A_{n-1}, for each n.

math.QA↗

arXiv2007

Finite-dimensional simple Poisson modules

We prove a result that can be applied to determine the finite-dimensional simple Poisson modules over a Poisson algebra and apply it to numerous examples. In the discussion of the examples, the emphasis is on the correspondence with the finite-dimensional simple modules over deformations and on the behaviour of finite-dimensional simple Poisson modules on the passage from a Poisson algebra to the Poisson subalgebra of invariants for the action of a finite group of Poisson automorphisms.

math.RA↗

arXiv2007

Reversible skew Laurent polynomial rings and deformations of Poisson automorphisms

A skew Laurent polynomial ring R[x^{\pm 1};α] is reversible if it has a reversing automorphism, that is, an automorphism of period two that transposes x and x^{-1} and restricts to an automorphism of R. We study invariants for reversing automorphisms and apply our methods to determine the rings of invariants of reversing automorphisms of two simple skew Laurent polynomial rings, namely a localization of the enveloping algebra of the two-dimensional non-abelian solvable Lie algebra and the coordinate ring of the quantum torus. These two skew Laurent polynomial rings are deformations of Poisson algebras and we interpret their reversing automorphisms and their invariants as deformations of Poisson automorphisms and their invariants.

math.RA↗