Root Systems, Cartan Matrix and Dynkin Diagrams in Classification of Lie Algebras

2018 ◽  
Vol 26 (3) ◽  
pp. 1-10
Author(s):  
Um Alemam ◽  
Mohamed Ahmed
2019 ◽  
Vol 19 (11) ◽  
pp. 2050223
Author(s):  
Noriaki Kamiya ◽  
Daniel Mondoc

In this work, we discuss a classification of [Formula: see text]-Freudenthal–Kantor triple systems defined by bilinear forms and give all examples of such triple systems. From these results, we may see a construction of some simple Lie algebras or superalgebras associated with their Freudenthal–Kantor triple systems. We also show that we can associate a complex structure into these ([Formula: see text]-Freudenthal–Kantor triple systems. Further, we introduce the concept of Dynkin diagrams associated to such [Formula: see text]-Freudenthal–Kantor triple systems and the corresponding Lie (super) algebra construction.


1982 ◽  
Vol 34 (1) ◽  
pp. 63-79 ◽  
Author(s):  
R. V. Moody ◽  
T. Yokonuma

This paper is concerned with two things. The first is a (primarily) geometric axiomatic description for the systems of real roots of Lie algebras arising from (generalized) Cartan matrices. The description is base free and is a natural extension of the well-known axiomatic description of finite root systems. The primary component of our description is an open convex cone which, following Looijenga [3], we call the Tits cone. In fact it was Looijenga's paper that led to this axiomatic formulation. Unlike his construction, the dimension of the Tits cone is not tightly connected to the dimension of the Cartan matrix which it eventually yields. This leads us to the second part of the paper which concerns the construction of Cartan matrices of low row rank. We can show that if we have an l × l Cartan matrix of row rank n, then we can model an axiomatic description of it with a cone of dimension n + 1.


2002 ◽  
Vol 45 (4) ◽  
pp. 711-731 ◽  
Author(s):  
Yoji Yoshii

AbstractQuantum tori with graded involution appear as coordinate algebras of extended affine Lie algebras of type A1, C and BC. We classify them in the category of algebras with involution. From this, we obtain precise information on the root systems of extended affine Lie algebras of type C.


Author(s):  
Florian Beck ◽  
Ron Donagi ◽  
Katrin Wendland

Abstract Folding of ADE-Dynkin diagrams according to graph automorphisms yields irreducible Dynkin diagrams of $\textrm{ABCDEFG}$-types. This folding procedure allows to trace back the properties of the corresponding simple Lie algebras or groups to those of $\textrm{ADE}$-type. In this article, we implement the techniques of folding by graph automorphisms for Hitchin integrable systems. We show that the fixed point loci of these automorphisms are isomorphic as algebraic integrable systems to the Hitchin systems of the folded groups away from singular fibers. The latter Hitchin systems are isomorphic to the intermediate Jacobian fibrations of Calabi–Yau orbifold stacks constructed by the 1st author. We construct simultaneous crepant resolutions of the associated singular quasi-projective Calabi–Yau three-folds and compare the resulting intermediate Jacobian fibrations to the corresponding Hitchin systems.


2009 ◽  
Vol 220 (1) ◽  
pp. 59-124 ◽  
Author(s):  
I. Heckenberger
Keyword(s):  

2016 ◽  
Vol 110 ◽  
pp. 25-29 ◽  
Author(s):  
Hamid Darabi ◽  
Farshid Saeedi ◽  
Mehdi Eshrati
Keyword(s):  

Author(s):  
D. F. Holt ◽  
N. Spaltenstein

AbstractThe classification of the nilpotent orbits in the Lie algebra of a reductive algebraic group (over an algebraically closed field) is given in all the cases where it was not previously known (E7 and E8 in bad characteristic, F4 in characteristic 3). The paper exploits the tight relation with the corresponding situation over a finite field. A computer is used to study this case for suitable choices of the finite field.


2017 ◽  
Vol 190 (1) ◽  
pp. 23-51 ◽  
Author(s):  
Kenro Furutani ◽  
Irina Markina

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