An algebraic scheme associated with the noncommutative KP hierarchy and some of its extensions
arXiv:nlin/0501003 · doi:10.1088/0305-4470/38/24/005
Abstract
A well-known ansatz (`trace method') for soliton solutions turns the equations of the (noncommutative) KP hierarchy, and those of certain extensions, into families of algebraic sum identities. We develop an algebraic formalism, in particular involving a (mixable) shuffle product, to explore their structure. More precisely, we show that the equations of the noncommutative KP hierarchy and its extension (xncKP) in the case of a Moyal-deformed product, as derived in previous work, correspond to identities in this algebra. Furthermore, the Moyal product is replaced by a more general associative product. This leads to a new even more general extension of the noncommutative KP hierarchy. Relations with Rota-Baxter algebras are established.
59 pages, relative to the second version a few minor corrections, but quite a lot of amendments, to appear in J. Phys. A
References in corpus (24)
- String Theory and Noncommutative Geometry
- Noncommutative Field Theory
- Loday-type Algebras and the Rota-Baxter Relation
- D-particles, Matrix Integrals and KP hierachy
- Quantum Bi-Hamiltonian Systems
- Integrable Renormalization II: the general case
- Spitzer's Identity and the Algebraic Birkhoff Decomposition in pQFT
- Hopf algebras, from basics to applications to renormalization
- Exact Noncommutative KP and KdV Multi-solitons
- An alternative approach to KP hierarchy in matrix models
- Nonabelian KP hierarchy with Moyal algebraic coefficients
- Extension of Noncommutative Soliton Hierarchies
- Factorization methods for Noncommutative KP and Toda hierarchy
- Renormalization in quantum field theory and the Riemann-Hilbert problem
- Integrable Renormalization I: the Ladder Case
- Explorations of the Extended ncKP Hierarchy
- On the associative Nijenhuis Relation
- New mathematical structures in renormalizable quantum field theories
- Witten's conjecture, Virasoro conjecture, and invariance of tautological equations
- Extension of Moyal-deformed hierarchies of soliton equations
- Noncommutative Dirac-Born-Infeld Action for D-brane
- The uses of Connes and Kreimer's algebraic formulation of renormalization theory
- Quasideterminants and Casimir elements for the general Lie superalgebra
- A Realization of Matrix KP Hierarchy by Coincident D-brane States
Cited by in corpus (20)
- On a direct approach to quasideterminant solutions of a noncommutative modified KP equation
- Quasideterminant solutions of a non-Abelian Toda lattice and kink solutions of a matrix sine-Gordon equation
- Noncommutative Ward's Conjecture and Integrable Systems
- Integrable Discretisations for a Class of Nonlinear Schrodinger Equations on Grassmann Algebras
- Grassmann extensions of Yang-Baxter maps
- Functional representations of integrable hierarchies
- Darboux transformations for a twisted derivation and quasideterminant solutions to the super KdV equation
- With a Cole-Hopf transformation to solutions of the noncommutative KP hierarchy in terms of Wronski matrices
- On Reductions of Noncommutative Anti-Self-Dual Yang-Mills Equations
- On solutions to the non-Abelian Hirota-Miwa equation and its continuum limits
- On the consistency of a Grassmann extended lattice Boussinesq system
- Nonassociativity and Integrable Hierarchies
- Soliton Solutions of Noncommutative Anti-Self-Dual Yang-Mills Equations
- A new approach to deformation equations of noncommutative KP hierarchies
- The non-commutative Korteweg--de Vries hierarchy and combinatorial Poppe algebra
- Darboux transformations, discrete integrable systems and related Yang-Baxter maps
- Algebraic identities associated with KP and AKNS hierarchies
- Quasideterminant solutions to a noncommutative -difference two-dimensional Toda lattice equation
- Darboux Transformations for (2+1)-Dimensional Extensions of the KP Hierarchy
- Quasi-symmetric functions and the KP hierarchy