Supersymmetric Toda Lattice Hierarchies

Kadyshevsky, V. G.; Сорин, А.С.

doi:10.1007/978-94-010-0720-7_10

Cited by 13 publications

(12 citation statements)

References 18 publications

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“…The 2D TL hierarchy was first studied in [1,2], and at present two different nontrivial supersymmetric extensions of 2D TL are known. They are the N = (2|2) [3]- [12] and N = (0|2) [12,9] supersymmetric TL hierarchies that possess a different number of supersymmetries and contain the N = (2|2) and N = (0|2) TL equations as subsystems. Quite recently, the 2D generalized fermionic TL equations have been introduced [11] and their two reductions related to the N = (2|2) and N = (0|2) supersymmetric TL equations were considered.…”

Section: Introductionmentioning

confidence: 99%

Fermionic one- and two-dimensional Toda lattice hierarchies and their bi-Hamiltonian structures

2005

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By exhibiting the corresponding Lax pair representations we propose a wide class of integrable two-dimensional (2D) fermionic Toda lattice (TL) hierarchies which includes the 2D N = (2|2) and N = (0|2) supersymmetric TL hierarchies as particular cases. Performing their reduction to the one-dimensional case by imposing suitable constraints we derive the corresponding 1D fermionic TL hierarchies. We develop the generalized graded R-matrix formalism using the generalized graded bracket on the space of graded operators with an involution generalizing the graded commutator in superalgebras, which allows one to describe these hierarchies in the framework of the Hamiltonian formalism and construct their first two Hamiltonian structures. The first Hamiltonian structure is obtained for both bosonic and fermionic Lax operators while the second Hamiltonian structure is established for bosonic Lax operators only. We propose the graded modified Yang-Baxter equation in the operator form and demonstrate that for the class of graded antisymmetric R-matrices it is equivalent to the tensor form of the graded classical Yang-Baxter equation.

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Section: Introductionmentioning

confidence: 99%

Fermionic one- and two-dimensional Toda lattice hierarchies and their bi-Hamiltonian structures

2005

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“…The reduction bj = 0 (2.20) of eqs. (2.15) leads to the 2D N = (0|2) supersymmetric Toda lattice equations [12,5]. One can easily see that fermionic symmetries (2.16) are not consistent with this reduction, while fermionic symmetries (2.17) are consistent and form the algebra of the N = (0|2) supersymmetry.…”

Section: D Generalized Fermionic Toda Lattice Equationsmentioning

confidence: 99%

Generalized fermionic discrete Toda hierarchy

Gribanov

Kadyshevsky

Сорин

2004

Discrete Dynamics in Nature and Society

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We describe bi-Hamiltonian structure and Lax-pair formulation with the spectral parameter of the generalized fermionic Toda lattice hierarchy as well as its bosonic and fermionic symmetries for different (including periodic) boundary conditions. Its two reductions-N = 4 and N = 2 supersymmetric Toda lattice hierarchies-in different (including canonical) bases are investigated. Its r-matrix description, monodromy matrix, and spectral curves are discussed.

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“…They are the N =(2|2) [2]- [7] and N =(0|2) [4], [7] supersymmetric TL hierarchies, which have different numbers of supersymmetries and contain the N =(2|2) and N =(0|2) TL equations as subsystems. Quite recently, the 2D generalized fermionic TL equations were introduced [6], and two reductions of these equations leading to the N =(2|2) and N =(0|2) supersymmetric TL equations were considered.…”

Section: Introductionmentioning

confidence: 99%

Hamiltonian Structures of Fermionic Two-Dimensional Toda Lattice Hierarchies

2006

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By exhibiting the corresponding Lax-pair representations, we propose a wide class of integrable twodimensional (2D) fermionic Toda lattice (TL) hierarchies, which includes the 2D N =(2|2) and N =(0|2) supersymmetric TL hierarchies as particular cases. We develop the generalized graded R-matrix formalism using the generalized graded bracket on the space of graded operators with involution generalizing the graded commutator in superalgebras, which allows describing these hierarchies in the framework of the Hamiltonian formalism and constructing their first two Hamiltonian structures. We obtain the first Hamiltonian structure for both bosonic and fermionic Lax operators and the second Hamiltonian structure only for bosonic Lax operators.

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Supersymmetric Toda Lattice Hierarchies

Cited by 13 publications

References 18 publications

Fermionic one- and two-dimensional Toda lattice hierarchies and their bi-Hamiltonian structures

Fermionic one- and two-dimensional Toda lattice hierarchies and their bi-Hamiltonian structures

Generalized fermionic discrete Toda hierarchy

Hamiltonian Structures of Fermionic Two-Dimensional Toda Lattice Hierarchies

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