1,595 research outputs found
The Classification of the Simply Laced Berger Graphs from Calabi-Yau spaces
The algebraic approach to the construction of the reflexive polyhedra that
yield Calabi-Yau spaces in three or more complex dimensions with K3 fibres
reveals graphs that include and generalize the Dynkin diagrams associated with
gauge symmetries. In this work we continue to study the structure of graphs
obtained from reflexive polyhedra. The objective is to describe the
``simply laced'' cases, those graphs obtained from three dimensional spaces
with K3 fibers which lead to symmetric matrices. We study both the affine and,
derived from them, non-affine cases. We present root and weight structurea for
them. We study in particular those graphs leading to generalizations of the
exceptional simply laced cases and . We show how
these integral matrices can be assigned: they may be obtained by relaxing the
restrictions on the individual entries of the generalized Cartan matrices
associated with the Dynkin diagrams that characterize Cartan-Lie and affine
Kac-Moody algebras. These graphs keep, however, the affine structure present in
Kac-Moody Dynkin diagrams. We conjecture that these generalized simply laced
graphs and associated link matrices may characterize generalizations of
Cartan-Lie and affine Kac-Moody algebras
Hunting for the New Symmetries in Calabi-Yau Jungles
It was proposed that the Calabi-Yau geometry can be intrinsically connected
with some new symmetries, some new algebras. In order to do this it has been
analyzed the graphs constructed from K3-fibre CY_d (d \geq 3) reflexive
polyhedra. The graphs can be naturally get in the frames of Universal
Calabi-Yau algebra (UCYA) and may be decode by universal way with changing of
some restrictions on the generalized Cartan matrices associated with the Dynkin
diagrams that characterize affine Kac-Moody algebras. We propose that these new
Berger graphs can be directly connected with the generalizations of Lie and
Kac-Moody algebras.Comment: 29 pages, 15 figure
Root systems from Toric Calabi-Yau Geometry. Towards new algebraic structures and symmetries in physics?
The algebraic approach to the construction of the reflexive polyhedra that
yield Calabi-Yau spaces in three or more complex dimensions with K3 fibres
reveals graphs that include and generalize the Dynkin diagrams associated with
gauge symmetries. In this work we continue to study the structure of graphs
obtained from reflexive polyhedra. We show how some particularly defined
integral matrices can be assigned to these diagrams. This family of matrices
and its associated graphs may be obtained by relaxing the restrictions on the
individual entries of the generalized Cartan matrices associated with the
Dynkin diagrams that characterize Cartan-Lie and affine Kac-Moody algebras.
These graphs keep however the affine structure, as it was in Kac-Moody Dynkin
diagrams. We presented a possible root structure for some simple cases. We
conjecture that these generalized graphs and associated link matrices may
characterize generalizations of these algebras.Comment: 24 pages, 6 figure
Polyhedral realization of Crystal bases for Generalized Kac-Moody Algebras
In this paper, we give polyhedral realization of the crystal of
for the generalized Kac-Moody algebras. As applications,
we give explicit descriptions of crystals for the generalized Kac-Moody
algebras of rank 2, 3 and {\it Monster Lie algebras}.Comment: 20 page
Valued Graphs and the Representation Theory of Lie Algebras
Quivers (directed graphs) and species (a generalization of quivers) and their
representations play a key role in many areas of mathematics including
combinatorics, geometry, and algebra. Their importance is especially apparent
in their applications to the representation theory of associative algebras, Lie
algebras, and quantum groups. In this paper, we discuss the most important
results in the representation theory of species, such as Dlab and Ringel's
extension of Gabriel's theorem, which classifies all species of finite and tame
representation type. We also explain the link between species and K-species
(where K is a field). Namely, we show that the category of K-species can be
viewed as a subcategory of the category of species. Furthermore, we prove two
results about the structure of the tensor ring of a species containing no
oriented cycles that do not appear in the literature. Specifically, we prove
that two such species have isomorphic tensor rings if and only if they are
isomorphic as "crushed" species, and we show that if K is a perfect field, then
the tensor algebra of a K-species tensored with the algebraic closure of K is
isomorphic to, or Morita equivalent to, the path algebra of a quiver.Comment: 36 page
- …