13,294 research outputs found
Degree 3 Cohomological Invariants of Groups that are Neither Simply Connected nor Adjoint
In a recent paper A. Merkurjev constructed an exact sequence which includes
as one of the terms the group of degree 3 normalized cohomological invariants
of a semisimple algebraic group G, greatly extending results of M. Rost for
simply connected quasi-simple groups. Furthermore, in the aforementioned paper,
Merkurjev uses his exact sequence to determine the groups of invariants for all
semisimple adjoint groups of inner type. The goal of this paper is to use
Merkurjev's sequence to compute the group of invariants for the remaining split
cases, namely groups of types A and D that are neither simply connected nor
adjoint. This description not only demonstrates the existence of many
previously unstudied invariants but also allows us to extend several known
results which relate these invariants to the Rost invariant and algebras with
involution
Complex solutions to Painleve IV equation through supersymmetric quantum mechanics
In this work, supersymmetric quantum mechanics will be used to obtain complex
solutions to Painleve IV equation with real parameters. We will also focus on
the properties of the associated Hamiltonians, i.e. the algebraic structure,
the eigenfunctions and the energy spectra.Comment: 5 pages, 3 figures. Talk given at the Advanced Summer School 2011,
Cinvestav (Mexico City), July 201
Supersymmetric Quantum Mechanics and Painlev\'e IV Equation
As it has been proven, the determination of general one-dimensional
Schr\"odinger Hamiltonians having third-order differential ladder operators
requires to solve the Painlev\'e IV equation. In this work, it will be shown
that some specific subsets of the higher-order supersymmetric partners of the
harmonic oscillator possess third-order differential ladder operators. This
allows us to introduce a simple technique for generating solutions of the
Painlev\'e IV equation. Finally, we classify these solutions into three
relevant hierarchies.Comment: Proceedings of the Workshop 'Supersymmetric Quantum Mechanics and
Spectral Design' (July 18-30, 2010, Benasque, Spain
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