We introduce a Frobenius algebra-valued Kadomtsev-Petviashvili (KP) hierarchy and show the existence of Frobenius algebra-valued τ-function for this hierarchy. In addition, we construct its Hamiltonian structures by using the Adler-Dickey-Gelfand method. As a byproduct of these constructions, we show that the coupled KP hierarchy, defined by Casati and Ortenzi [J. Geom. Phys. 56, 418-449 (2006)], has at least n-“basic” different local bi-Hamiltonian structures. Finally, via the construction of the second Hamiltonian structures, we obtain some local matrix, or Frobenius algebra-valued, generalizations of classical W-algebras.
REFERENCES
1.
M.
Adler
, “On a trace functional for formal pseudodifferential operators and the symplectic structure of the Korteweg-de Vries type equations
,” Invent. Math.
50
, 219
–248
(1979
).2.
I.
Bakas
, “Higher spin fields and the Gelfand-Dickey algebra
,” Commun. Math. Phys.
123
, 627
–639
(1989
).3.
A.
Bilal
, “Non-local matrix generalizations of W-algebras
,” Commun. Math. Phys.
170
, 117
–150
(1995
).4.
P.
Casati
and G.
Ortenzi
, “New integrable hierarchies from vertex operator representations of polynomial Lie algebras
,” J. Geom. Phys.
56
, 418
–449
(2006
).5.
E.
Date
, M.
Jimbo
, M.
Kashiwara
, and T.
Miwa
, “Transformation groups for soliton equations
,” in Proceedings of R.I.M.S. Symposium on Nonlinear Integrable Systems–Classical Theory and Quantum Theory
(World Scientific
, 1983
), pp. 39
–119
.6.
L. A.
Dickey
, “On Hamiltonian and Lagrangian formalisms for the KP hierarchy of integrable equations
,” Ann. N. Y. Acad. Sci.
491
, 131
–148
(1987
).7.
L. A.
Dickey
, “Lectures on classical W-algebras
,” Acta Appl. Math.
47
, 243
–321
(1997
).8.
L. A.
Dickey
, Soliton Equations and Hamiltonian Systems
, Advanced Series in Mathematical Physics
Vol. 26
, 2nd ed. (World Scientific
, 2003
).9.
B.
Dubrovin
, Geometry of 2D Topological Field Theories
, Lecture Notes in Mathematics
Vol. 1620
(Springer
, 1996
), pp. 120
–348
.10.
P.
Di Franscessco
, C.
Itzykson
, and J. B.
Zuber
, “Classical W-algebras
,” Commun. Math. Phys.
140
, 543
–567
(1991
).11.
A. P.
Fordy
, A. G.
Reyman
, and M. A.
Semenov-Tian-Shansky
, “Classical r-matrices and compatible Poisson brackets for coupled KdV systems
,” Lett. Math. Phys.
17
, 25
–29
(1989
).12.
I. M.
Gelfand
and L. A.
Dickey
, “Family of Hamiltonian structures connected with integrable non-linear equations
,” preprint, IPM, Moscow (1978
) (in Russian)[English version in
I. M.
Gelfand
and L. A.
Dickey
, Collected Papers of I. M. Gelfand
(Springer-Verlag
, 1987
), Vol. 1
, pp. 625
–646
].13.
R.
Hirota
, X. B.
Hu
, and X. Y.
Tang
, “A vector potential KdV equation and vector Ito equation: Soliton solutions, bilinear Bäcklund transformations and Lax pairs.
,” J. Math. Anal. Appl.
288
(1
), 326
–348
(2003
).14.
B. A.
Kupershmidt
, KP or mKP: Noncommutative Mathematics of Lagrangian, Hamiltonian, and Integrable Systems
, Mathematical Surveys and Monographs
(American Mathematical Society
, 2000
), Vol. 78
.15.
J.
van de Leur
, “Bäcklund transformations for new integrable hierarchies related to the polynomial Lie algebra
,” J. Geom. Phys.
57
, 435
–447
(2007
).16.
W. X.
Ma
and B.
Fuchssteinery
, “The bi-Hamiltonian structure of the perturbation equations of KdV hierarchy
,” Phys. Lett. A
213
, 49
–55
(1996
).17.
P. J.
Olver
and V. V.
Sokolov
, “Integrable evolution equations on associative algebras
,” Commun. Math. Phys.
2
, 245
–268
(1998
).18.
A. O.
Radul
, “Two series of Hamiltonian structures for the hierarchy of Kadomtsev-Petviashvili equations
,” in Applied Methods of Nonlinear Analysis and Control
, edited by Mironov
, Moroz
, and Tshernyatin
(Moscow State University
, Moscow
, 1987
), pp. 149
–157
.19.
M.
Sato
, “Soliton equations as dynamical systems on infinite dimensional Grassmann manifolds
,” RIMS Kokyuroku
439
, 30
–46
(1981
).20.
I. A. B.
Strachan
, “The Moyal bracket and the dispersionless limit of the KP hierarchy
,” J. Phys. A: Math. Gen.
28
, 1967
–1975
(1995
).21.
I. A. B.
Strachan
and D.
Zuo
, “Frobenius manifolds and Frobenius algebra-valued integrable systems
,” preprint arXiv:1403.0021.22.
Y.
Watanabe
, “Hamiltonian structure of Sato’s hierarchy of KP equations and a coadjoint orbit of a certain formal Lie group
,” Lett. Math. Phys.
7
(2
), 99
–106
(1983
).23.
D.
Zuo
, “On the Kuperschmidt-Wilson theorem for the Moyal-Kadomtsev-Petviasfvili hierarchy
,” Inverse Probl.
22
, 1959
–1966
(2006
).24.
25.
D.
Zuo
, “The Frobenius-Virasoro algebra and Euler equations
,” J. Geom. Phys.
86
, 203
–210
(2014
).26.
H.
Zhang
and D.
Zuo
, “Hamiltonian structures of the constrained -valued KP hierarchy
,” Rep. Math. Phys.
76
, 116
–129
(2015
).27.
K-L.
Tian
and D.
Zuo
, “Free field realizations of some local matrix W-algebras
” (unpublished).28.
A skew mapping is said to be Hamiltonian if (1) is a Lie subalgebra and (2) the 2-form ω defined by is closed.
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