Loop Algebra Research Articles

A generalized Zakharov–Shabat equation with finite-band solutions and a soliton-equation hierarchy with an arbitrary parameter

In this paper, a generalized Zakharov–Shabat equation ( g-ZS equation), which is an isospectral problem, is introduced by using a loop algebra G ∼ . From the stationary zero curvature equation we define the Lenard gradients { g j } and the corresponding generalized AKNS ( g-AKNS) vector fields { X j } and X k flows. Employing the nonlinearization method, we obtain the generalized Zhakharov–Shabat Bargmann ( g-ZS-B) system and prove that it is Liouville integrable by introducing elliptic coordinates and evolution equations. The explicit relations of the X k flows and the polynomial integrals { H k } are established. Finally, we obtain the finite-band solutions of the g-ZS equation via the Abel–Jacobian coordinates. In addition, a soliton hierarchy and its Hamiltonian structure with an arbitrary parameter k are derived.

The n × n KdV hierarchy

We introduce two new soliton hierarchies that are generalizations of the KdV hierarchy. Our hierarchies are restrictions of the AKNS n × n hierarchy coming from two unusual splittings of the loop algebra. These splittings come from automorphisms of the loop algebra instead of automorphisms of \({sl (n, \mathbb{C})}\) . The flows in the hierarchy include systems of coupled nonlinear Schrodinger equations. Since they are constructed from a Lie algebra splitting, the general method gives formal inverse scattering, bi-Hamiltonian structures, commuting flows, and Backlund transformations for these hierarchies.

TWO ENLARGED LOOP ALGEBRAS FOR OBTAINING NEW INTEGRABLE HIERARCHIES

Taking a loop algebra [Formula: see text] we obtain an integrable soliton hierarchy which is similar to the well-known Kaup–Newell (KN) hierarchy, but it is not. We call it a modified KN (mKN) hierarchy. Then two new enlarged loop algebras of the loop algebra [Formula: see text] are established, respectively, which are used to establish isospectral problems. Thus, two various types of integrable soliton-equation hierarchies along with multi-component potential functions are obtained. Their Hamiltonian structures are also obtained by the variational identity. The second hierarchy is integrable couplings of the mKN hierarchy. This paper provides a clue for generating loop algebras, specially, gives an approach for producing new integrable systems. If we obtain a new soliton hierarchy, we could deduce its symmetries, conserved laws, Darboux transformations, soliton solutions and so on. Hence, the way presented in the paper is an important aspect to obtain new integrable systems in soliton theory.

New extended Lie algebra and the generalized integrable Liouville hierarchy

Abstract Starting from a new Lie algebra G 1 , the corresponding loop algebra G 1 ∼ is presented, from which a Liouville integrable hierarchy is given by using of variational identity. It follows that an expanding Lie algebra G 2 is obtained based on G 1 , furthermore, a related Lax integrable hierarchy is presented by making use of its related loop algebra G 2 ∼ . With the help of variational identity, it is not difficult to prove that the hierarchy has Hamiltonian structure and is Liouville integrable. It is also can be seen that the second hierarchy is the expanding model of the first one.

New DLW Hierarchy of an Integrable Coupling and Its Hamiltonian Structure

A type of higher-dimensional loop algebra is constructed from which an isospectral problem is established. It follows that an integrable coupling, actually an extended integrable model of the existed solitary hierarchy of equations, is obtained by taking use of the zero curvature equation, whose Hamiltonian structure is worked out by employing the constructed quadratic identity.

Root subsystems of loop extensions

We completely classify the real root subsystems of root systems of loop algebras of Kac–Moody Lie algebras. This classification involves new notions of “admissible subgroups” of the coweight lattice of a root system Ψ, and “scaling functions” on Ψ. Our results generalise and simplify earlier work on subsystems of real affine root systems.

A NEW METHOD FOR GENERATING A NEW INTEGRABLE SYSTEM

With the help of the known Lie algebra given by Zhang,2a new higher-dimensional Lie algebra G is obtained by generalizing the commutative operations in the Lie algebras. Using a subalgebra [Formula: see text] of a loop algebra [Formula: see text] which corresponds to the Lie algebra G, a new heat-conduction equation hierarchy with some constrained conditions, is obtained. We again consider the constrained conditions as new evolution equations, the new scheme for generating soliton equations are given. Then we use the loop algebra [Formula: see text] to further establish an isospectral problem and derive an extending integrable model of the above heat-condition hierarchy, we also obtain a corresponding extending constrained condition which is thought as a type of evolution equations.

Generating infinite-dimensional algebras from loop algebras by expanding Maurer–Cartan forms

It is shown that the expansion methods developed in de Azcarraga et al.[Nucl. Phys. B 662, 185 (2003); Class. Quantum Grav. 21, S1375 (2004)]; can be generalized so that they permit to study the expansion of algebras of loops, both when the compact finite-dimensional algebra and the algebra of loops have a decomposition into two subspaces.

Ground State Representations of Loop Algebras

Let g be a simple Lie algebra, Lg be the loop algebra of g. Fixing a point in S^1 and identifying the real line with the punctured circle, we consider the subalgebra Sg of Lg of rapidly decreasing elements on R. We classify the translation-invariant 2-cocycles on Sg. We show that the ground state representation of Sg is unique for each cocycle. These ground states correspond precisely to the vacuum representations of Lg.

Quantum gravity coupled to matter via noncommutative geometry

We show that the principal part of the Dirac Hamiltonian in 3+1 dimensions emerges in a semi-classical approximation from a construction which encodes the kinematics of quantum gravity. The construction is a spectral triple over a configuration space of connections. It involves an algebra of holonomy loops represented as bounded operators on a separable Hilbert space and a Dirac-type operator. Semi-classical states, which involve an averaging over points at which the product between loops is defined, are constructed and it is shown that the Dirac Hamiltonian emerges as the expectation value of the Dirac-type operator on these states in a semi-classical approximation.

A Few Expanding Integrable Models, Hamiltonian Structures and Constrained Flows

Two kinds of higher-dimensional Lie algebras and their loop algebras are introduced, for which a few expanding integrable models including the coupling integrable couplings of the Broer—Kaup (BK) hierarchy and the dispersive long wave (DLW) hierarchy as well as the TB hierarchy are obtained. From the reductions of the coupling integrable couplings, the corresponding coupled integrable couplings of the BK equation, the DLW equation, and the TB equation are obtained, respectively. Especially, the coupling integrable coupling of the TB equation reduces to a few integrable couplings of the well-known mKdV equation. The Hamiltonian structures of the coupling integrable couplings of the three kinds of soliton hierarchies are worked out, respectively, by employing the variational identity. Finally, we decompose the BK hierarchy of evolution equations into x-constrained flows and tn-constrained flows whose adjoint representations and the Lax pairs are given.

Two Types of Coupling Integrable Couplings of the S-mKdV Hierarchy

Firstly 4 Lie algebras are constructed. Then applications of the loop algebra are presented to obtain two types of coupling integrable couplings of the S-mKdV hierarchy by using Tu scheme. The coupling integrable couplings of the S-mKdV hierarchy obtained in the paper reduce to the coupling integrable couplings of the mKdV equation and the coupling integrable couplings of the nonlinear Schrödinger equation respectively. The method given in the paper can be used to other hierarchies generally.

Nonlinear continuous integrable Hamiltonian couplings

Based on a kind of special non-semisimple Lie algebras, a scheme is presented for constructing nonlinear continuous integrable couplings. Variational identities over the corresponding loop algebras are used to furnish Hamiltonian structures for the resulting continuous integrable couplings. The application of the scheme is illustrated by an example of nonlinear continuous integrable Hamiltonian couplings of the AKNS hierarchy of soliton equations.

Double Integrable Couplings and Their Constructing Method

To extend the study scopes of integrable couplings, the notion of double integrable couplings is proposed in the paper. The zero curvature equation appearing in the constructing method built in the paper consists of the elements of a new loop algebra which is obtained by using perturbation method. Therefore, the approach given in the paper has extensive applicable values, that is, it applies to investigate a lot of double integrable couplings of the known integrable hierarchies of evolution equations. As for explicit applications of the method proposed in the paper, the double integrable couplings of the AKNS hierarchy and the KN hierarchy are worked out, respectively.

Quiver varieties and cluster algebras

Motivated by a recent conjecture by Hernandez and Leclerc, we embed a Fomin-Zelevinsky cluster algebra into the Grothendieck ring R of the category of representations of quantum loop algebras Uq(Lg) of a symmetric Kac-Moody Lie algebra, studied earlier by the author via perverse sheaves on graded quiver varieties. Graded quiver varieties controlling the image can be identified with varieties which Lusztig used to define the canonical base. The cluster monomials form a subset of the base given by the classes of simple modules in R, or Lusztig’s dual canonical base. The conjectures that cluster monomials are positive and linearly independent (and probably many other conjectures) of Fomin and Zelevinsky follow as consequences when there is a seed with a bipartite quiver. Simple modules corresponding to cluster monomials factorize into tensor products of “prime” simple ones according to the cluster expansion.

The Classical R-Matrix of AdS/CFT and its Lie Dialgebra Structure

The classical integrable structure of Z_4-graded supercoset sigma-models, arising in the AdS/CFT correspondence, is formulated within the R-matrix approach. The central object in this construction is the standard R-matrix of the Z_4-twisted loop algebra. However, in order to correctly describe the Lax matrix within this formalism, the standard inner product on this twisted loop algebra requires a further twist induced by the Zhukovsky map, which also plays a key role in the AdS/CFT correspondence. The non-ultralocality of the sigma-model can be understood as stemming from this latter twist since it leads to a non skew-symmetric R-matrix.

Two Types of Expanding Lie Algebra and New Expanding Integrable Systems

From a new Lie algebra proposed by Zhang, two expanding Lie algebras and its corresponding loop algebras are obtained. Two expanding integrable systems are produced with the help of the generalized zero curvature equation. One of them has complex Hamiltion structure with the help of generalized Tu formula (GTM).

Quantum loop subalgebra and eigenvectors of the superintegrable chiral Potts transfer matrices

It has been shown in earlier works that for Q = 0 and L a multiple of N, the ground state sector eigenspace of the superintegrable τ2(tq) model is highly degenerate and is generated by a quantum loop algebra . Furthermore, this loop algebra can be decomposed into r = (N − 1)L/N simple algebras. For Q ≠ 0, we shall show here that the corresponding eigenspace of τ2(tq) is still highly degenerate, but splits into two spaces, each containing 2r − 1 independent eigenvectors. The generators for the subalgebras, and also for the quantum loop subalgebra, are given generalizing those in the Q = 0 case. However, the Serre relations for the generators of the loop subalgebra are only proven for some states, tested on small systems and conjectured otherwise. Assuming their validity we construct the eigenvectors of the Q ≠ 0 ground state sectors for the transfer matrix of the superintegrable chiral Potts model.

A Class of New Lie Algebra, the Corresponding g-mKdV Hierarchy and Its Hamiltonian Structure

A class of new Lie algebra B(3) is constructed, which is far different from the known Lie algebra A(n-1). Based on the corresponding loop algebra (B) over bar (3), the generalized mKdV hierarchy is established. In order to look for the Hamiltonian structure of such integrable system, a generalized trace functional of matrices is introduced, whose special case is just the well-known trace identity. Finally, its expanding integrable model is worked out by use of an enlarged Lie algebra.

Real forms of complex surfaces of constant mean curvature

It is known that complex constant mean curvature (CMC for short) immersions in C 3 are natural complexifications of CMC-immersions in R 3 . In this paper, conversely we consider real form surfaces of a complex CMC-immersion, which are defined from real forms of the twisted sl(2, C) loop algebra Asl(2, ℂ) σ , and classify all such surfaces according to the classification of real forms of Asl(2,ℂ) σ . There are seven classes of surfaces, which are called integrable surfaces, and all integrable surfaces will be characterized by the (Lorentz) harmonicities of their Gauss maps into the symmetric spaces S 2 , H 2 , S 1,1 or the 4-symmetric space SL(2,ℂ)/U(1). We also give a unification to all integrable surfaces via the generalized Weierstrass type representation.