Fixed-point-free automorphisms of Lie algebras

1989 ◽  
Vol 5 (1) ◽  
pp. 95-96 ◽  
Author(s):  
Zha Jianguo
Keyword(s):  
Fixed Point ◽  
Lie Algebras

scholarly journals Folding of Hitchin Systems and Crepant Resolutions

2021 ◽  
Author(s):  
Florian Beck ◽  
Ron Donagi ◽  
Katrin Wendland
Keyword(s):  
Fixed Point ◽  
Lie Algebras ◽  
Integrable Systems ◽  
Trace Back ◽  
Dynkin Diagrams ◽  
Or Groups ◽  
Hitchin Systems ◽  
Singular Fibers ◽  
Crepant Resolutions ◽  
Intermediate Jacobian

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.

scholarly journals Fixed point subalgebras of extended affine Lie algebras

2005 ◽  
Vol 287 (2) ◽  
pp. 351-380 ◽  
Author(s):  
Saeid Azam ◽  
Stephen Berman ◽  
Malihe Yousofzadeh
Keyword(s):  
Fixed Point ◽  
Lie Algebras ◽  
Affine Lie Algebras

scholarly journals Approximate -Lie Homomorphisms and Jordan -Lie Homomorphisms on -Lie Algebras

2012 ◽  
Vol 2012 ◽  
pp. 1-11
Author(s):  
M. Eshaghi Gordji ◽  
G. H. Kim
Keyword(s):  
Functional Equation ◽  
Fixed Point ◽  
Lie Algebras ◽  
Jensen Functional Equation ◽  
Jensen Functional ◽  
Fixed Point Methods ◽  
The Stability

Using fixed point methods, we establish the stability of -Lie homomorphisms and Jordan -Lie homomorphisms on -Lie algebras associated to the following generalized Jensen functional equation .

Lie Tori and Their Fixed Point Subalgebras

2009 ◽  
Vol 16 (03) ◽  
pp. 381-396 ◽  
Author(s):  
Saeid Azam ◽  
Valiollah Khalili
Keyword(s):  
Fixed Point ◽  
Fixed Points ◽  
Lie Algebras ◽  
Finite Order ◽  
Affine Lie Algebras ◽  
Order Automorphism

We study the fixed point subalgebra of a centerless irreducible Lie torus under a certain finite order automorphism. We investigate which axioms of a Lie torus hold for the fixed points and which do not. We relate our study to some recent results about the fixed points of extended affine Lie algebras under a class of finite order automorphisms.

Generalized Reductive Lie Algebras: Connections With Extended Affine Lie Algebras and Lie Tori

2006 ◽  
Vol 58 (2) ◽  
pp. 225-248 ◽  
Author(s):  
Saeid Azam
Keyword(s):  
Fixed Point ◽  
Root System ◽  
Lie Algebra ◽  
Lie Algebras ◽  
Affine Lie Algebras ◽  
The Core ◽  
Direct Sum ◽  
Reductive Lie Algebra ◽  
Intensive Investigation

AbstractWe investigate a class of Lie algebras which we call generalized reductive Lie algebras. These are generalizations of semi-simple, reductive, and affine Kac–Moody Lie algebras. A generalized reductive Lie algebra which has an irreducible root system is said to be irreducible and we note that this class of algebras has been under intensive investigation in recent years. They have also been called extended affine Lie algebras. The larger class of generalized reductive Lie algebras has not been so intensively investigated. We study themin this paper and note that one way they arise is as fixed point subalgebras of finite order automorphisms. We show that the core modulo the center of a generalized reductive Lie algebra is a direct sum of centerless Lie tori. Therefore one can use the results known about the classification of centerless Lie tori to classify the cores modulo centers of generalized reductive Lie algebras.

scholarly journals Cyclotomic Discriminantal Arrangements and Diagram Automorphisms of Lie Algebras

2017 ◽  
Vol 2019 (11) ◽  
pp. 3376-3458 ◽  
Author(s):  
Alexander Varchenko ◽  
Charles Young
Keyword(s):  
Fixed Point ◽  
Lie Algebras ◽  
Hyperplane Arrangements ◽  
Poincaré Polynomial ◽  
Chain Complexes ◽  
Affine Hyperplane ◽  
Diagram Automorphisms ◽  
Gaudin Models ◽  
Nilpotent Subalgebras ◽  
Flag Complexes

Abstract We identify a class of affine hyperplane arrangements that we call cyclotomic discriminantal arrangements. We establish correspondences between the flag and Aomoto complexes of such arrangements and chain complexes for nilpotent subalgebras of Kac–Moody type Lie algebras with diagram automorphisms. As part of this construction, we find that flag complexes naturally give rise to a certain cocycle on the fixed-point subalgebras of such diagram automorphisms. As a byproduct, we show that the Bethe vectors of cyclotomic Gaudin models associated to diagram automorphisms are nonzero. We also obtain the Poincare polynomial for the cyclotomic discriminantal arrangements.

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