UNIVERSITY OF KENT
School of Mathematics, Statistics and Actuarial Science
Mathematics Seminar Programme
Spring Term 2010
15 January
Title:
Carl Bender (Washington University in St. Louis)
Making sense of non-Hermitian Hamiltonians
Abstract: The average quantum physicist on the street believes that a quantum-mechanical
Hamiltonian must be Dirac Hermitian (invariant under combined matrix transposition and
complex conjugation) in order to guarantee that the energy eigenvalues are real and that time
evolution is unitary. However, the Hamiltonian $H=p^2+ix^3$, which is obviously not Dirac
Hermitian, has a real positive discrete spectrum and generates unitary time evolution, and
thus it defines a fully consistent and physical quantum theory.
Evidently, the axiom of Dirac Hermiticity is too restrictive. While $H=p^2+ix^3$ is not
Dirac Hermitian, it is PT symmetric; that is, invariant under combined space reflection P and
time reversal T. The quantum mechanics defined by a PT-symmetric Hamiltonian is a
complex generalization of ordinary quantum mechanics. When quantum mechanics is
extended into the complex domain, new kinds of theories having strange and remarkable
properties emerge. Some of these properties have recently been verified in laboratory
experiments. If one generalizes classical mechanics into the complex domain, the resulting
theories have equally remarkable properties.
[This talk will be presented at an elementary colloquium-style level and will be easy to
understand and broadly accessible.]
Kent-Porto Day
19 February Paula Carvalho (Porto)
Title:
Down-Up algebras and their representations
Abstract: A class of algebras called down-up algebras was introduced by G. Benkart and T.
Roby in 1998 (J.Algebra 209, 305-344) as a generalization of algebras generated by a pair of
operators acting on the vector space spanned over the complex numbers freely by the
elements of a partially ordered set. Given parameters $\alpha$, $\beta$, $\gamma$ in a field
$K$, the down-up algebra $A(\alpha, \beta,\gamma)$ is the associative $K$-algebra defined
by two generators $u$ and $d$ and two cubic relations that depend upon the parameters.
In the first part of the talk we will look at some ring theoretical properties of this class of
algebras. In the second part we study finiteness conditions on injective hulls of simple
modules over Noetherian down-up algebras.
19 February Christian Lomp (Porto)
Title:
Ring theoretical aspects of weak Hopf action
Abstract: Groupoid actions are typical examples of weak Hopf algebra actions that are not
ordinary Hopf actions. Looking at actions as certain operators acting on an algebra, we
extend known result on Hopf actions to this setting. In particular we get new insight into the
existence of non-trivial central invariant elements in non-trivial H-stable ideals of subdirect
products of certain H-prime module algebras satisfying a polynomial identity by considering
an adapted version of Kaplansky's theorem and by introducing a Brown-McCoy radical for
H-module algebras. We extend Puczylowski and Smoktunowicz's description of the BrownMcCoy radical of a polynomial ring to module algebras and apply our result to left
bialgebroid measurings, gradings and involutions. Following Linchenko and Montgomery's
arguments we show that the smash product of a semiprime module algebra, satisfying a
polynomial identity and an involutive weak Hopf algebra is semiprime.
19 February Samuel Lopes (Porto)
Title:
Automorphisms of certain generalised Weyl algebras
Abstract: We will introduce generalised down-up algebras, as defined by Cassidy and Shelton.
As the name indicates, they naturally generalise the down-up algebras of Benkart and Roby,
whose definition was motivated by algebraic properties of the down and up operators of
certain posets. These algebras, when Noetherian, are part of a wider class introduced by
Bavula, the so-called Generalised Weyl Algebras (GWA). Automorphism groups of GWA's
were studied by Bavula and Jordan in the case that the base ring is a polynomial algebra in
one variable. In this joint work with Paula Carvalho, we determine the automorphism groups
of conformal Noetherian generalised down up algebras, under some restrictions on the
parameters. These provide instances of automorphism groups of GWA's in the case that the
base ring is a polynomial algebra in two variables.
2 March
Title:
Mike Keane (Wesleyan College Connecticut / Leiden)
The binomial transformation.
Abstract: This mapping from the unit interval to itself is also known as the Pascal
transformation, and has a simple definition using cutting and stacking of intervals. In the
lecture I can give an interesting argument for its ergodicity, based on the Lebesgue density
theorem and recurrence of one-dimensional random walk, and discuss the question, still open,
as to whether it is mixing - even weak mixing is unknown.
16 March
Title:
David Berman (Queen Mary)
Interacting Branes in M-theory
Abstract: This is an introductory review on M-theory, its branes and recent developments in
the interacting theory of branes.