Quasiclassical quantization of the periodic Toda chain from the point of view of Lie algebras

1983 ◽  
Vol 54 (3) ◽  
pp. 312-314 ◽  
Author(s):  
Yu. M. Vorob'ev ◽  
S. Yu. Dobrokhotov
Keyword(s):  
Lie Algebras ◽  
Toda Chain ◽  
Point Of View ◽  
Periodic Toda Chain

Lecture on SUSY Gauge Theories and Integrable Systems

1997 ◽  
Vol 11 (26n27) ◽  
pp. 3093-3124
Author(s):  
A. Marshakov
Keyword(s):  
Field Theory ◽  
Quantum Field Theory ◽  
Integrable Systems ◽  
Gauge Theories ◽  
Toda Chain ◽  
Quantum Field ◽  
Standard Quantum ◽  
Integrable Equations ◽  
Complex Curves ◽  
Periodic Toda Chain

I consider main features of the formulation of the finite-gap solutions to integrable equations in terms of complex curves and generating 1-differential. The example of periodic Toda chain solutions is considered in detail. Recently found exact nonperturbative solutions to [Formula: see text] SUSY gauge theories are formulated using the methods of the theory of integrable systems and where possible the parallels between standard quantum field theory results and solutions to the integrable systems are discussed.

Representations Subduced on an Ideal of a Lie Algebra

1962 ◽  
Vol 14 ◽  
pp. 293-303 ◽  
Author(s):  
B. Noonan
Keyword(s):  
Irreducible Representation ◽  
Lie Algebra ◽  
Lie Algebras ◽  
Kronecker Product ◽  
Point Of View ◽  
Irreducible Representations ◽  
Closed Field ◽  
Algebraically Closed Field ◽  
Representations Of Lie Algebras ◽  
The Ideal

This paper considers the properties of the representation of a Lie algebra when restricted to an ideal, the subduced* representation of the ideal. This point of view leads to new forms for irreducible representations of Lie algebras, once the concept of matrices of invariance is developed. This concept permits us to show that irreducible representations of a Lie algebra, over an algebraically closed field, can be expressed as a Lie-Kronecker product whose factors are associated with the representation subduced on an ideal. Conversely, if one has such factors, it is shown that they can be put together to give an irreducible representation of the Lie algebra. A valuable guide to this work was supplied by a paper of Clifford (1).

scholarly journals Singularities, Lax degeneracies and Maslov indices of the periodic Toda chain

Nonlinearity ◽  
2005 ◽  
Vol 18 (6) ◽  
pp. 2795-2813 ◽  
Author(s):  
J A Foxman ◽  
J M Robbins
Keyword(s):  
Toda Chain ◽  
Periodic Toda Chain

Chaos upon soliton decay in a perturbed periodic toda chain

1986 ◽  
Vol 23 (1-3) ◽  
pp. 374-380 ◽  
Author(s):  
Karlheinz Geist ◽  
Werner Lauterborn
Keyword(s):  
Toda Chain ◽  
Perturbed Periodic ◽  
Periodic Toda Chain

scholarly journals ON LIE INDUCTION AND THE EXCEPTIONAL SERIES

2005 ◽  
Vol 04 (06) ◽  
pp. 707-737 ◽  
Author(s):  
JAN E. GRABOWSKI
Keyword(s):  
Lie Algebras ◽  
Necessary Conditions ◽  
Point Of View ◽  
Lie Bialgebras ◽  
Simple Complex ◽  
Single Node ◽  
Finite Dimensional ◽  
Dynkin Diagrams ◽  
Induction Process ◽  
And Algebra

Lie bialgebras occur as the principal objects in the infinitesimalization of the theory of quantum groups — the semi-classical theory. Their relationship with the quantum theory has made available some new tools that we can apply to classical questions. In this paper, we study the simple complex Lie algebras using the double-bosonization construction of Majid. This construction expresses algebraically the induction process given by adding and removing nodes in Dynkin diagrams, which we call Lie induction. We first analyze the deletion of nodes, corresponding to the restriction of adjoint representations to subalgebras. This uses a natural grading associated to each node. We give explicit calculations of the module and algebra structures in the case of the deletion of a single node from the Dynkin diagram for a simple Lie (bi-)algebra. We next consider the inverse process, namely that of adding nodes, and give some necessary conditions for the simplicity of the induced algebra. Finally, we apply these to the exceptional series of simple Lie algebras, in the context of finding obstructions to the existence of finite-dimensional simple complex algebras of types E9, F5 and G3. In particular, our methods give a new point of view on why there cannot exist such an algebra of type E9.

scholarly journals Duality Identities for Moduli Functions of Generalized Melvin Solutions Related to Classical Lie Algebras of Rank 4

2018 ◽  
Vol 2018 ◽  
pp. 1-10 ◽  
Author(s):  
S. V. Bolokhov ◽  
V. D. Ivashchuk
Keyword(s):  
Lie Algebras ◽  
Toda Chain ◽  
Scalar Fields ◽  
Cartan Matrix ◽  
Radial Coordinate ◽  
Cylindrically Symmetric ◽  
Symmetry Properties ◽  
The Matrix ◽  
Chain Type ◽  
Internal Symmetries

We consider generalized Melvin-like solutions associated with nonexceptional Lie algebras of rank 4 (namely, A4, B4, C4, and D4) corresponding to certain internal symmetries of the solutions. The system under consideration is a static cylindrically symmetric gravitational configuration in D dimensions in presence of four Abelian 2-forms and four scalar fields. The solution is governed by four moduli functions Hs(z) (s=1,…,4) of squared radial coordinate z=ρ2 obeying four differential equations of the Toda chain type. These functions turn out to be polynomials of powers (n1,n2,n3,n4)=(4,6,6,4),(8,14,18,10),(7,12,15,16),(6,10,6,6) for Lie algebras A4, B4, C4, and D4, respectively. The asymptotic behaviour for the polynomials at large distances is governed by some integer-valued 4×4 matrix ν connected in a certain way with the inverse Cartan matrix of the Lie algebra and (in A4 case) the matrix representing a generator of the Z2-group of symmetry of the Dynkin diagram. The symmetry properties and duality identities for polynomials are obtained, as well as asymptotic relations for solutions at large distances. We also calculate 2-form flux integrals over 2-dimensional discs and corresponding Wilson loop factors over their boundaries.

scholarly journals Quantization conditions for the periodic Toda chain: Inadequacy of Bethe-ansatz methods

1989 ◽  
Vol 39 (16) ◽  
pp. 11800-11809 ◽  
Author(s):  
Michael Fowler ◽  
Holger Frahm
Keyword(s):  
Bethe Ansatz ◽  
Toda Chain ◽  
Periodic Toda Chain

Semiclassical quantization of the periodic Toda chain

1993 ◽  
Vol 26 (24) ◽  
pp. 7589-7613 ◽  
Author(s):  
F Gohmann ◽  
W Pesch ◽  
F G Mertens
Keyword(s):  
Toda Chain ◽  
Semiclassical Quantization ◽  
Periodic Toda Chain

EXTENDED STATE EXPANSIONS AND THE UNIVERSALITY OF WITTEN’S VERSION OF LINK POLYNOMIAL THEORY

1990 ◽  
Vol 05 (10) ◽  
pp. 1975-2003 ◽  
Author(s):  
MO-LIN GE ◽  
YU-QUAN LI ◽  
KANG XUE
Keyword(s):  
Lie Algebras ◽  
Braid Group ◽  
Point Of View ◽  
Extended State ◽  
Explicit Representations ◽  
Markov Trace ◽  
Chern Simons ◽  
Polynomial Theory

The Witten’s version for constructing the skein relations of link polynomials based on (2+1) Chern-Simons Lagrangian is shown to be universal for Lie algebras. The extended state calculations are developed to give the explicit representations of braid group and new understanding of framing factor from the point of view of generalized Markov trace.

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