7 research outputs found
Higher dimensional Automorphic Lie Algebras
The paper presents the complete classification of Automorphic Lie Algebras
based on , where the symmetry group is finite
and the orbit is any of the exceptional -orbits in .
A key feature of the classification is the study of the algebras in the context
of classical invariant theory. This provides on one hand a powerful tool from
the computational point of view, on the other it opens new questions from an
algebraic perspective, which suggest further applications of these algebras,
beyond the context of integrable systems. In particular, the research shows
that Automorphic Lie Algebras associated to the
groups (tetrahedral, octahedral and
icosahedral groups) depend on the group through the automorphic functions only,
thus they are group independent as Lie algebras. This can be established by
defining a Chevalley normal form for these algebras, generalising this
classical notion to the case of Lie algebras over a polynomial ring.Comment: 43 pages, standard LaTeX2
Automorphic Lie Algebras with dihedral symmetry
The concept of Automorphic Lie Algebras arises in the context of reduction
groups introduced in the early 1980s in the field of integrable systems.
Automorphic Lie Algebras are obtained by imposing a discrete group symmetry on
a current algebra of Krichever-Novikov type. Past work shows remarkable
uniformity between algebras associated to different reduction groups. For
example, if the base Lie algebra is and the poles
of the Automorphic Lie Algebra are restricted to an exceptional orbit of the
symmetry group, changing the reduction group does not affect the Lie algebra
structure. In the present research we fix the reduction group to be the
dihedral group and vary the orbit of poles as well as the group action on the
base Lie algebra. We find a uniform description of Automorphic Lie Algebras
with dihedral symmetry, valid for poles at exceptional and generic orbits.Comment: 20 pages, 5 tables, standard LaTeX2
Automorphic Lie Algebras and Cohomology of Root Systems
A cohomology theory of root systems emerges naturally in the context of
Automorphic Lie Algebras, where it helps formulating some structure theory
questions. In particular, one can find concrete models for an Automorphic Lie
Algebra by integrating cocycles. In this paper we define this cohomology and
show its connection with the theory of Automorphic Lie Algebras. Furthermore,
we discuss its properties: we define the cup product, we show that it can be
restricted to symmetric forms, that it is equivariant with respect to the
automorphism group of the root system, and finally we show acyclicity at
dimension two of the symmetric part, which is exactly what is needed to find
concrete models for Automorphic Lie Algebras.
Furthermore, we show how the cohomology of root systems finds application
beyond the theory of Automorphic Lie Algebras by applying it to the theory of
contractions and filtrations of Lie algebras. In particular, we show that
contractions associated to Cartan -filtrations of simple Lie
algebras are classified by -cocycles, due again to the vanishing of the
symmetric part of the second cohomology group.Comment: 26 pages, standard LaTeX2
Polyhedral Groups in
We classify embeddings of the finite groups , and in the Lie
group up to conjugation.Comment: 6 pages. To appear in the Glasgow Mathematical Journa
Riemann-Hilbert Problem, Integrability and Reductions
Abstract. The present paper is dedicated to integrable models with Mikhailov reduction groups GR ≃ Dh. Their Lax representation allows us to prove, that their solution is equivalent to solving Riemann-Hilbert problems, whose contours depend on the realization of the GR-action on the spectral parameter. Two new examples of Nonlinear Evolution Equations (NLEE) with Dh symmetries are presented
Riemann-Hilbert problem, integrability and reductions
The present paper is dedicated to integrable models with Mikhailov reduction
groups Their Lax representation allows us to prove,
that their solution is equivalent to solving Riemann-Hilbert problems, whose
contours depend on the realization of the -action on the spectral
parameter. Two new examples of Nonlinear Evolution Equations (NLEE) with
symmetries are presented.Comment: 19 pages, 3 figures, Dedicated to Darryl Holm's 70th birthda