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.

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.

On Lie Algebras with Many Nilpotent Subalgebras

2006 ◽  
Vol 34 (2) ◽  
pp. 595-600 ◽  
Author(s):  
Alexander A. Lashkhi ◽  
Irene Zimmermann
Keyword(s):  
Lie Algebras ◽  
Nilpotent Subalgebras

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 Vertex Lie algebras and cyclotomic coinvariants

2017 ◽  
Vol 19 (02) ◽  
pp. 1650015 ◽  
Author(s):  
Benoît Vicedo ◽  
Charles Young
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Bethe Ansatz ◽  
Riemann Sphere ◽  
Marked Point ◽  
Vertex Algebra ◽  
The Other ◽  
Gaudin Models ◽  
Equivariant Functions ◽  
Marked Points

Given a vertex Lie algebra [Formula: see text] equipped with an action by automorphisms of a cyclic group [Formula: see text], we define spaces of cyclotomic coinvariants over the Riemann sphere. These are quotients of tensor products of smooth modules over “local” Lie algebras [Formula: see text] assigned to marked points [Formula: see text], by the action of a “global” Lie algebra [Formula: see text] of [Formula: see text]-equivariant functions. On the other hand, the universal enveloping vertex algebra [Formula: see text] of [Formula: see text] is itself a vertex Lie algebra with an induced action of [Formula: see text]. This gives “big” analogs of the Lie algebras above. From these we construct the space of “big” cyclotomic coinvariants, i.e. coinvariants with respect to [Formula: see text]. We prove that these two definitions of cyclotomic coinvariants in fact coincide, provided the origin is included as a marked point. As a corollary, we prove a result on the functoriality of cyclotomic coinvariants which we require for the solution of cyclotomic Gaudin models in [B. Vicedo and C. Young, Cyclotomic Gaudin models: Construction and Bethe ansatz, preprint (2014); arXiv:1409.6937]. At the origin, which is fixed by [Formula: see text], one must assign a module over the stable subalgebra [Formula: see text] of [Formula: see text]. This module becomes a [Formula: see text]-quasi-module in the sense of Li. As a bi-product we obtain an iterate formula for such quasi-modules.

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

CHIRAL ALGEBRAS—TWO EXAMPLES

1991 ◽  
Vol 06 (30) ◽  
pp. 5481-5494
Author(s):  
RON COHEN
Keyword(s):  
Fixed Point ◽  
Automorphism Group ◽  
Duality Relation ◽  
Poincaré Polynomial ◽  
Duality Relations ◽  
Duality Map ◽  
Superconformal Theories ◽  
Scalar Theories ◽  
Chiral Algebras

We examine the chiral algebras of quotient superconformal theories by studying two examples: the SO(n) HSS models and the models based on the SU (n) group. We study the automorphism group which identifies fields in each model, and its fixed-point fields. We calculate the Poincare polynomial for the SU (n) example, see whether there are duality relations and find new Landau-Ginzburg scalar theories. For the second example we explain how to calculate the number of fields in the algebra and show a duality relation by giving the duality map explicitly.

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 .

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