Dynkin diagrams

1970 ◽  
pp. 34-43
Author(s):  
Ian Stewart
Keyword(s):  
Dynkin Diagrams

scholarly journals 2, 12, 117, 1959, 45171, 1170086, …: a Hilbert series for the QCD chiral Lagrangian

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Lukáš Gráf ◽  
Brian Henning ◽  
Xiaochuan Lu ◽  
Tom Melia ◽  
Hitoshi Murayama
Keyword(s):  
Hilbert Series ◽  
Charge Conjugation ◽  
External Fields ◽  
Dynkin Diagrams ◽  
Non Linear ◽  
Very High ◽  
Chiral Lagrangian

Abstract We apply Hilbert series techniques to the enumeration of operators in the mesonic QCD chiral Lagrangian. Existing Hilbert series technologies for non-linear realizations are extended to incorporate the external fields. The action of charge conjugation is addressed by folding the $$ \mathfrak{su}(n) $$ su n Dynkin diagrams, which we detail in an appendix that can be read separately as it has potential broader applications. New results include the enumeration of anomalous operators appearing in the chiral Lagrangian at order p8, as well as enumeration of CP-even, CP-odd, C-odd, and P-odd terms beginning from order p6. The method is extendable to very high orders, and we present results up to order p16.(The title sequence is the number of independent C-even and P-even operators in the mesonic QCD chiral Lagrangian with three light flavors of quarks, at chiral dimensions p2, p4, p6, …)

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.

EXT ALGEBRA OF NICHOLS ALGEBRAS OF CARTAN TYPE A2

2013 ◽  
Vol 12 (04) ◽  
pp. 1250191
Author(s):  
XIAOLAN YU ◽  
YINHUO ZHANG
Keyword(s):  
Spectral Sequence ◽  
Hopf Algebras ◽  
Type A ◽  
Serre Spectral Sequence ◽  
Nichols Algebra ◽  
Cartan Type ◽  
Dynkin Diagrams ◽  
Full Structure ◽  
Pointed Hopf Algebras ◽  
Nichols Algebras

We give the full structure of the Ext algebra of any Nichols algebra of Cartan type A2 by using the Hochschild–Serre spectral sequence. As an application, we show that the pointed Hopf algebras [Formula: see text] with Dynkin diagrams of type A, D, or E, except for A1 and A1 × A1 with the order NJ > 2 for at least one component J, are wild.

Automorphisms, isomorphisms, and Tits classification

2015 ◽  
Author(s):  
Brian Conrad ◽  
Gopal Prasad
Keyword(s):  
Affine Group ◽  
Classification Theorem ◽  
Isomorphism Theorem ◽  
Reductive Groups ◽  
Semisimple Groups ◽  
Dynkin Diagrams ◽  
Weil Restriction ◽  
Characteristic 2 ◽  
Smooth Affine

This chapter considers automorphisms, isomorphisms, and Tits classification. It begins by establishing a version of the Isomorphism Theorem for pseudo-split pseudo-reductive groups, along with a pseudo-reductive variant of the Isogeny Theorem for split connected semisimple groups. The key to both proofs is a technique to construct pseudo-reductive subgroups of an ambient smooth affine group. Some instructive examples over imperfect fields k of characteristic 2 are given. The chapter goes on to discuss the behavior of the k-group ZG,C with respect to Weil restriction in the pseudoreductive case. It also describes automorphism schemes for pseudo-reductive groups, focusing only on the pseudo-semisimple case because commutative pseudo-reductive groups that are not tori generally have a non-representable automorphism functor. Finally, it examines Tits-style classification, using Dynkin diagrams to express the classification theorem.

scholarly journals Quantum Computing, Seifert Surfaces, and Singular Fibers

2019 ◽  
Vol 1 (1) ◽  
pp. 12-22 ◽  
Author(s):  
Michel Planat ◽  
Raymond Aschheim ◽  
Marcelo M. Amaral ◽  
Klee Irwin
Keyword(s):  
Quantum Computing ◽  
Quantum Computer ◽  
Complete Information ◽  
Singular Fiber ◽  
Dynkin Diagrams ◽  
Singular Fibers ◽  
Seifert Surfaces ◽  
Trefoil Knot ◽  
Universal Quantum

The fundamental group π 1 ( L ) of a knot or link L may be used to generate magic states appropriate for performing universal quantum computation and simultaneously for retrieving complete information about the processed quantum states. In this paper, one defines braids whose closure is the L of such a quantum computer model and computes their braid-induced Seifert surfaces and the corresponding Alexander polynomial. In particular, some d-fold coverings of the trefoil knot, with d = 3 , 4, 6, or 12, define appropriate links L, and the latter two cases connect to the Dynkin diagrams of E 6 and D 4 , respectively. In this new context, one finds that this correspondence continues with Kodaira’s classification of elliptic singular fibers. The Seifert fibered toroidal manifold Σ ′ , at the boundary of the singular fiber E 8 ˜ , allows possible models of quantum computing.

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