Geodesic
Quasi-Frobenius Algebras and Their Integrable $N$-Parameter Deformations Generated by Compatible $(N\times N)$ Metrics of Constant Riemannian Curvature
Teoretičeskaâ i matematičeskaâ fizika, Tome 136 (2003) no. 1, pp. 20-29 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

We prove that the equations describing compatible $(N\times N)$ metrics of constant Riemannian curvature define a special class of integrable $N$-parameter deformations of quasi-Frobenius (in general, noncommutative) algebras. We discuss connections with open-closed two-dimensional topological field theories, associativity equations, and Frobenius and quasi-Frobenius manifolds. We conjecture that open-closed two-dimensional topological field theories correspond to a special class of integrable deformations of associative quasi-Frobenius algebras.
Mots-clés : quasi-Frobenius algebra
Keywords: Frobenius algebra, integrable deformation of an algebra, topological field theory, compatible metrics, constant-curvature metrics, integrable system, quasi-Frobenius manifold, Frobenius manifold, flat pencil of metrics, associativity equations. 
@article{TMF_2003_136_1_a1,
     author = {O. I. Mokhov},
     title = {Quasi-Frobenius {Algebras} and {Their} {Integrable} $N${-Parameter} {Deformations} {Generated} by {Compatible} $(N\times N)$ {Metrics} of {Constant} {Riemannian} {Curvature}},
     journal = {Teoreti\v{c}eska\^a i matemati\v{c}eska\^a fizika},
     pages = {20--29},
     year = {2003},
     volume = {136},
     number = {1},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TMF_2003_136_1_a1/}
}
TY  - JOUR
AU  - O. I. Mokhov
TI  - Quasi-Frobenius Algebras and Their Integrable $N$-Parameter Deformations Generated by Compatible $(N\times N)$ Metrics of Constant Riemannian Curvature
JO  - Teoretičeskaâ i matematičeskaâ fizika
PY  - 2003
SP  - 20
EP  - 29
VL  - 136
IS  - 1
UR  - http://geodesic.mathdoc.fr/item/TMF_2003_136_1_a1/
LA  - ru
ID  - TMF_2003_136_1_a1
ER  - 
%0 Journal Article
%A O. I. Mokhov
%T Quasi-Frobenius Algebras and Their Integrable $N$-Parameter Deformations Generated by Compatible $(N\times N)$ Metrics of Constant Riemannian Curvature
%J Teoretičeskaâ i matematičeskaâ fizika
%D 2003
%P 20-29
%V 136
%N 1
%U http://geodesic.mathdoc.fr/item/TMF_2003_136_1_a1/
%G ru
%F TMF_2003_136_1_a1
O. I. Mokhov. Quasi-Frobenius Algebras and Their Integrable $N$-Parameter Deformations Generated by Compatible $(N\times N)$ Metrics of Constant Riemannian Curvature. Teoretičeskaâ i matematičeskaâ fizika, Tome 136 (2003) no. 1, pp. 20-29. http://geodesic.mathdoc.fr/item/TMF_2003_136_1_a1/

[1] I. M. Gelfand, I. Ya. Dorfman, Funkts. analiz i ego prilozh., 13:4 (1979), 13–30 | MR | Zbl

[2] I. M. Gelfand, I. Ya. Dorfman, Funkts. analiz i ego prilozh., 15:3 (1981), 23–40 | MR | Zbl

[3] I. Dorfman, Dirac Structures and Integrability of Nonlinear Evolution Equations, Wiley, Chichester, 1993 | MR

[4] A. A. Balinskii, S. P. Novikov, DAN SSSR, 283:5 (1985), 1036–1039 | MR | Zbl

[5] S. P. Novikov, UMN, 40:4 (1985), 79–89 | MR

[6] E. I. Zelmanov, DAN SSSR, 292:6 (1987), 1294–1297 | MR | Zbl

[7] E. B. Vinberg, DAN SSSR, 141:3 (1961), 521–524 | MR | Zbl

[8] O. I. Mokhov, Funkts. analiz i ego prilozh., 22:4 (1988), 92–93 | MR | Zbl

[9] B. Dubrovin, “Geometry of 2D topological field theories”, Integrable Systems and Quantum Groups, Lectures given at the 1st Session of the Centro Internazionale Matematico Estivo (C.I.M.E) (Montecatini Terme, Italy, June 14–22, 1993), Lecture Notes in Math., 1620, eds. M. Francaviglia, S. Greco, Springer, Berlin, 1996, 120–348 ; E-print hep-th/9407018 | DOI | MR | Zbl

[10] B. Dubrovin, Y. Zhang, Normal forms of hierarchies of integrable PDEs, Frobenius manifolds and Gromov–Witten invariants, Preprint SISSA 65/2001/FM, SISSA, Trieste, 2001; E-print math.DG/0108160

[11] O. I. Mokhov, TMF, 133:2 (2002), 279–288 ; E-print math.DG/0201281 | DOI | MR

[12] O. I. Mokhov, UMN, 56:6 (2001), 161–162 | DOI | MR | Zbl

[13] B. Dubrovin, Flat pencils of metrics and Frobenius manifolds, Preprint SISSA 25/98/FM, SISSA, Trieste, 1998 ; E-print math.DG/9803106 | MR

[14] O. I. Mokhov, TMF, 130:2 (2002), 233–250 ; E-print math.DG/0005081 | DOI | MR | Zbl

[15] O. I. Mokhov, UMN, 56:2 (2001), 221–222 | DOI | MR | Zbl

[16] O. I. Mokhov, UMN, 57:1 (2002), 157–158 | DOI | MR

[17] O. I. Mokhov, Funkts. analiz i ego prilozh., 35:2 (2001), 24–36 ; E-print math.DG/0005051 | DOI | MR | Zbl

[18] E. V. Ferapontov, J. Phys. A, 34 (2001), 2377–2388 ; E-print math.DG/0005221 | DOI | MR | Zbl

[19] S. M. Natanzon, Extended cohomological fields theory and noncommutative Frobenius manifolds, E-print math-ph/0206033 | MR

[20] A. Alexeevski, S. Natanzon, Non-commutative extensions of two-dimensional topological field theories and Hurwitz numbers for real algebraic curves, E-print math.GT/0202164 | MR

[21] O. I. Mokhov, UMN, 55:4 (2000), 217–218 | DOI | MR | Zbl

[22] B. A. Dubrovin, S. P. Novikov, DAN SSSR, 270:4 (1983), 781–785 | MR | Zbl

[23] O. I. Mokhov, E. V. Ferapontov, UMN, 45:3 (1990), 191–192 | MR | Zbl

[24] F. Magri, J. Math. Phys., 19:5 (1978), 1156–1162 | DOI | MR | Zbl

[25] O. I. Mokhov, Funkts. analiz i ego prilozh., 36:3 (2002), 36–47 ; E-print math.DG/0201280 | DOI | MR | Zbl

[26] O. I. Mokhov, UMN, 57:3 (2002), 155–156 | DOI | MR | Zbl

[27] O. I. Mokhov, Funkts. analiz i ego prilozh., 37:2 (2003) (to appear) ; E-print math.DG/0201223 | MR | Zbl

[28] O. I. Mokhov, UMN, 57:5 (2002), 157–158 | DOI | MR

[29] O. I. Mokhov, Phys. Lett. A, 166:3–4 (1992), 215–216 | DOI | MR

[30] O. I. Mokhov, UMN, 53:3 (1998), 85–192 | DOI | MR | Zbl

[31] O. I. Mokhov, Rev. Math. Math. Phys., 11:2 (2001), 1–204 | MR

[32] O. I. Mokhov, UMN, 52:6 (1997), 171–172 | DOI | MR

[33] O. I. Mokhov, Tr. MIRAN, 225, 1999, 284–300 | MR | Zbl

[34] O. I. Mokhov, Rep. Math. Phys., 43:1/2 (1999), 247–256 | DOI | MR | Zbl

[35] E. Witten, Nucl. Phys. B, 340 (1990), 281–332 | DOI | MR

[36] E. Witten, Surv. Diff. Geom., 1 (1991), 243–310 | DOI | MR

[37] R. Dijkgraaf, H. Verlinde, E. Verlinde, Nucl. Phys. B, 352 (1991), 59–86 | DOI | MR

[38] B. Dubrovin, Nucl. Phys. B, 379 (1992), 627–689 | DOI | MR