Super-trace Identity Research Articles

Nonlinear super integrable Hamiltonian couplings

Based on a kind of new Lie superalgebras, we proposed a scheme for constructing nonlinear super integrable couplings. Super trace identity over the corresponding loop superalgebras is used to furnish the super Hamiltonian structures for the resulting nonlinear super integrable couplings. We illustrate the application of the scheme by means of an enlarged super Ablowitz-Kaup-Newell-Segur (AKNS) matrix spectral problem and present a nonlinear super integrable Hamiltonian couplings for the super AKNS hierarchy.

Conservation laws and self-consistent sources for a super integrable equation hierarchy

In this paper, a super integrable equation hierarchy is considered based on a Lie superalgebra and supertrace identity. Then, a super integrable equation hierarchy with self-consistent sources is established. Furthermore, we introduce two variables F and G to construct conservation laws of the super integrable equation hierarchy and the first two conserved densities and fluxes are listed. It would be specially mentioned that the Fermi variables play an important role in super integrable systems which is different from the ordinary integrable systems.

Super-KN Hierarchy and Its Super-Hamiltonian Structure

Based on the basis of the constructed Lie super algebra, the super-isospectral problem of KN hierarchy is considered. Under the frame of the zero curvature equation, the super-KN hierarchy is obtained. Furthermore, its super-Hamiltonian structure is presented by using super-trace identity and it has super-bi-Hamiltonian structure.

A Super Extension of Kaup—Newell Hierarchy

With the help of the zero-curvature equation and the super trace identity, we derive a super extension of the Kaup—Newell hierarchy associated with a 3 × 3 matrix spectral problem and establish its super bi-Hamiltonian structures. Furthermore, infinite conservation laws of the super Kaup—Newell equation are obtained by using spectral parameter expansions.

The super-classical-Boussinesq hierarchy and its super-Hamiltonian structure

Based on the constructed Lie superalgebra, the super-classical-Boussinesq hierarchy is obtained. Then, its super-Hamiltonian structure is obtained by making use of super-trace identity. Furthermore, the super-classical-Boussinesq hierarchy is also integrable in the sense of Liouville.

Two super-integrable hierarchies and their super-Hamiltonian structures

Two Lie super algebras are constructed from which we establish two super-isospectral problems. Under the frame of the zero curvature equations, the super-GJ hierarchy and the super-Yang hierarchy are presented respectively. Meanwhile, their super-Hamiltonian structures are obtained by using super-trace identity.

Lie Algebra and Lie Super Algebra for Integrable Couplings of C-KdV Hierarchy

Based on the constructed Lie algebra and Lie super algebra, the integrable couplings and super-integrable couplings of the C-KdV hierarchy are obtained respectively. Furthermore, its super-Hamiltonian structures are presented by using super-trace identity.

A new super-extension of the KdV hierarchy

A new super-extension of the KdV hierarchy is proposed, which is associated with a 3×3 matrix spectral problem. Using super-trace identity, generalized bi-Hamiltonian structures of this hierarchy are established. Moreover, infinite conservation laws of the new super-KdV equation are derived.

A new eight-dimensional Lie superalgebra and two corresponding hierarchies of evolution equations

A new eight-dimensional Lie superalgebra is constructed and two isospectral problems with six potentials are designed. Corresponding hierarchies of nonlinear evolution equations, as well as super-AKNS and super-Levi, are derived. Their super-Hamiltonian structures are established by making use of the supertrace identity, and they are integrable in the sense of Liouville.

Hamiltonian and Super-Hamiltonian Extensions Related to Broer-Kaup-Kupershmidt System

Based on the Lie algebra A1, the integrable Broer-Kaup-Kupershmidt (BKK) system is revisited. The bi-Hamiltonian structure is constructed by the trace identity. Two extensions of the Lie algebra A1 are considered, i.e., the non-semi-simple Lie algebra of 4×4 matrix and the super-Lie algebra of 3×3 matrix, from which two hierarchies of soliton equations related to BKK system are given. With the aid of the generalized trace identity and the super-trace identity, the Hamiltonian and super-Hamiltonian structures of the resulting systems are constructed.

A LIE SUPERALGEBRA AND CORRESPONDING HIERARCHY OF EVOLUTION EQUATIONS

A Lie super algebra G is constructed and an isospectral problem with five potentials is designed. From there, a corresponding hierarchy of nonlinear evolution equations is derived and super-AKNS, super-TD, super-D-AKNS are deduced. Their super-Hamiltonian structure is established by making use of the supertrace identity, and they are intergrable in sense of Liouville.

TWO SUPER-INTEGRABLE SYSTEMS AND ASSOCIATED SUPER-HAMILTONIAN STRUCTURES

The super extensions of g-cKdV and mKdV integrable systems are proposed. Two hierarchies of super-integrable nonlinear evolution equations are found. In addition, making use of the super-trace identity, we construct the super-Hamiltonian structures of zero-curvature equations associated with Lie superalgebras.

SUPER-BURGERS SOLITON HIERARCHY AND ITS SUPER-HAMILTONIAN STRUCTURE

A super-Burgers hierarchy and its super-Hamiltonian structure is obtained respectively based on Lie super-algebra and is associated with super-trace identity.

SUPER-DISPERSIVE LONG WAVE HIERARCHY AND ITS SUPER-HAMILTONIAN STRUCTURES

Super-dispersive long wave hierarchy is obtained by use of the Lie super-algebra B(0, 1), then the super-Hamiltonian structure of the above system is given by the associated supertrace identity. The method can be used to produce the super-Hamiltonian structures of the other superintegrable systems.

Two types of Lie super-algebra for the super-integrable Tu-hierarchy and its super-Hamiltonian structure

Two different Lie super-algebras are constructed which establish two isospectral problems. Under the frame of the zero curvature equations, the corresponding super-integrable hierarchies of the Tu-hierarchy are obtained. By making use of the super-trace identity, the super-Hamiltonian structures of the above integrable hierarchies are generated, which are Liouville integrable.

Hamiltonian and super-Hamiltonian systems of a hierarchy of soliton equations

By considering an isospectral eigenvalue problem, a hierarchy of soliton equations are derived. Two types of extensions are presented by enlarging the associated spectral problem. With the aid of generalized trace identity and the super-trace identity, the Hamiltonian and super-Hamiltonian structures for the integrable extensions are established.

A Super-Soliton Hierarchy and Its Super-Hamiltonian Structure

A super-soliton hierarchy and its super-Hamiltonian structure is obtained respectively based on Lie super-algebra and associated super-trace identity.

Lie algebras and Lie super algebra for the integrable couplings of NLS–MKdV hierarchy

Lie algebras and Lie super algebra are constructed and integrable couplings of NLS–MKdV hierarchy are obtained. Furthermore, its Hamiltonian and Super-Hamiltonian are presented by using of quadric-form identity and super-trace identity. The method can be used to produce the Hamiltonian structures of the other integrable and super-integrable systems.

A supertrace identity and its applications to superintegrable systems

A supertrace identity on Lie superalgebras is established. It provides a tool for constructing super-Hamiltonian structures of zero curvature equations associated with Lie superalgebras. Applications in the case of the Lie superalgebra B(0,1) present super-Hamiltonian structures of a super-AKNS soliton hierarchy and a super-Dirac soliton hierarchy.

Central charge and renormalization in supersymmetric theories with vortices

Some quantum features of vortices in supersymmetric theories in 1+2 dimensions are studied in a manifestly supersymmetric setting of the superfield formalism. A close examination of the supercurrent that accommodates the central charge and super-Poincare charges in a supermultiplet reveals that there is no genuine quantum anomaly in the supertrace identity and in the supercharge algebra, with the central-charge operator given by the bare Fayet-Iliopoulos term alone. The central charge and the vortex spectrum undergo renormalization on taking the expectation value of the central-charge operator. It is shown that the vortex spectrum is exactly determined at one loop while the spectrum of the elementary excitations receives higher-order corrections.