15 research outputs found
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
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
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
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
Invariants of automorphic lie algebras
Automorphic Lie Algebras arise in the context of reduction groups introduced in the late 1970s [35] in the field of integrable systems. They are subalgebras of Lie algebras over a ring of rational functions, denied by invariance under the action of a finite group, the reduction group. Since their introduction in 2005 [29, 31], mathematicians aimed to classify Automorphic Lie Algebras. Past work shows remarkable uniformity between the Lie algebras associated to different reduction groups. That is, many Automorphic Lie Algebras with nonisomorphic reduction groups are isomorphic [4, 30]. In this thesis we set out to find the origin of these observations by searching for properties that are independent of the reduction group, called invariants of Automorphic Lie Algebras.
The uniformity of Automorphic Lie Algebras with nonisomorphic reduction groups starts at the Riemann sphere containing the spectral parameter, restricting the finite groups to the polyhedral groups. Through the use of classical invariant theory and the properties of this class of groups it is shown that Automorphic Lie Algebras are freely generated modules over the polynomial ring in one variable. Moreover, the number of generators equals the dimension of the base Lie algebra, yielding an invariant. This allows the definition of the determinant of invariant vectors which will turn out to be another invariant. A surprisingly simple formula is given expressing this determinant as a monomial in ground forms.
All invariants are used to set up a structure theory for Automorphic Lie Algebras. This naturally leads to a cohomology theory for root systems. A first exploration of this structure theory narrows down the search for Automorphic Lie Algebras signicantly. Various particular cases are fully determined by their invariants, including most of the previously studied Automorphic Lie Algebras, thereby providing an explanation for their uniformity.In addition, the structure theory advances the classification project. For example, it clarifies the effect of a change in pole orbit resulting in various new Cartan-Weyl normal form generators for Automorphic Lie Algebras. From a more general perspective, the success of the structure theory and root system cohomology in absence of a field promises interesting theoretical developments for Lie algebras over a graded ring
Hereditary automorphic Lie algebras
We show that automorphic Lie algebras which contain a Cartan subalgebra with a constant-spectrum, called hereditary, are completely described by 2-cocycles on a classical root system taking only two different values. This observation suggests a novel approach to their classification. By determining the values of the cocycles on opposite roots, we obtain the Killing form and the abelianization of the automorphic Lie algebra. The results are obtained by studying equivariant vectors on the projective line. As a byproduct, we describe a method to reduce the computation of the infinite-dimensional space of said equivariant vectors to a finite-dimensional linear computation and the determination of the ring of automorphic functions on the projective line
Polyhedral groups in (C)
We classify embeddings of the finite groups A4, S4 and A5 in the Lie group G2(C) up to conjugation