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1991 | 55 | 1 | 245-268
Tytuł artykułu

Supercomplex structures, surface soliton equations, and quasiconformal mappings

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Języki publikacji
EN
Abstrakty
EN
Hurwitz pairs and triples are discussed in connection with algebra, complex analysis, and field theory. The following results are obtained: (i) A field operator of Dirac type, which is called a Hurwitz operator, is introduced by use of a Hurwitz pair and its characterization is given (Theorem 1). (ii) A field equation of the elliptic Neveu-Schwarz model of superstring theory is obtained from the Hurwitz pair (𝔼⁴,𝔼³) (Theorem 2), and its counterpart connected with the Hurwitz triple $(𝔼^{11},𝔼^{11},𝔼^{26})$ is mentioned. (iii) Isospectral deformations of the Hurwitz operator of the Hurwitz pair (𝔼²,𝔼²) induce various soliton equations (Theorem 3). (iv) A special complex structure, which is called a supercomplex structure, is introduced on separable Hilbert spaces (Definition 10). A correspondence between such structures and reduction solutions of Sato's version of Kadomtsev-Petviashvili system is established (Theorem 4). (v) The general class of quasiconformal mappings in the plane is obtained from generalized Hurwitz pairs (Theorem 5). From these results we conclude that Hurwitz pairs and triples give rise to several interesting applications.
Słowa kluczowe
Rocznik
Tom
55
Numer
1
Strony
245-268
Opis fizyczny
Daty
wydano
1991
otrzymano
1990-09-14
Twórcy
  • Institute of Mathematics, Polish Academy of Sciences, Łódź Branch, Narutowicza 56, 90-136 Łódź, Poland
  • Institute of Mathematics, Polish Academy of Sciences, Łódź Branch, Narutowicza 56, 90-136 Łódź, Poland
autor
  • Department of Mathematics, College of Humanities And Sciences, Nihon University, Tokyo 156, Japan
Bibliografia
  • [1] M. J. Ablowitz, D. J. Kaup, A. C. Newell and H. Seguer, The inverse scattering transform-Fourier analysis for nonlinear problems, Stud. Appl. Math. 53 (1974), 245-315.
  • [2] J. Adem and J. Ławrynowicz, Construction of normed maps $ℝ^{11} × ℝ^{11} → ℝ^{26}$, preprint, Dep. de Mat., Centro de Investigación y de Estudios Avanzados, México 1990.
  • [3] J. Adem, J. Ławrynowicz and J. Rembieliński, Generalized Hurwitz maps of the type S × V → W, preprint, Dep. de Mat., Centro de Investigación y de Estudios Avanzados, México 1990.
  • [4] M. F. Atiyah, Geometry of Yang-Mills Fields, Lezioni Fermione, Academia Nazionale dei Lincei-Scuola Normale Superiore, Pisa 1979.
  • [5] E. Date, M. Jimbo, M. Kashiwara and T. Miwa, Transformation groups for soliton equations, in: Proc. RIMS Symp. Nonlinear Integrable Systems-Classical and Quantum Theory, Kyoto 1981, World Scientific, Singapore 1983, 39-119.
  • [6] R. Fueter, Über die analytische Darstellung der regulären Funktionen einer Quaternionenvariablen, Comment. Math. Helv. 8 (1936), 371-381.
  • [7] I. Fukuroya, S. Kanemaki and O. Suzuki, Hermitian Hurwitz pairs, preprint, Nihon Univ., Dept. of Math., 1988, 26 pp.
  • [8] M. B. Green, J.H. Schwarz and E. Witten, Superstring Theory I-II, Cambridge Monographs Math. Phys., Cambridge Univ. Press, Cambridge 1987.
  • [9] A. Hurwitz, Über die Komposition der quadratischen Formen von beliebig vielen Variablen, Nachr. Königl. Gesell. Wiss. Göttingen Math.-Phys. Kl. 1898, 309-316; reprinted in : A. Hurwitz, Mathematische Werke II, Birkhäuser, Basel 1933, 565-571.
  • [10] A. Hurwitz, Über die Komposition der quadratischen Formen, Math. Ann. 88 (1923), 1-25; reprinted in: A. Hurwitz, Mathematische Werke II, Birkhäuser, Basel 1933, 641-666.
  • [11] J. Kalina, J. Ławrynowicz and O. Suzuki, A field equation defined by a Hurwitz pair, Proc. 13th Winter School on Abstract Analysis, Srní (Bohemian Forest), Rend. Circ. Mat. Palermo (2) Suppl. 9 (1985), 117-128.
  • [12] J. Kalina, J. Ławrynowicz and O. Suzuki, Partial differential equations connected with some Clifford structures and the related quasiconformal mappings, Rend. Sem. Fac. Sci. Univ. Cagliari 57 (1987), 131-142.
  • [13] S. Kanemaki, Hurwitz pairs and octonions, in: Deformations of Mathematical Structures. Complex Analysis and Physical Applications, Selected papers from the Seminar on Deformations, Łódź-Lublin 1985/87, J. Ławrynowicz (ed.), Kluwer, Dordrecht 1989, 215-223.
  • [14] S. Kanemaki and O. Suzuki, Hermitian pre-Hurwitz pairs and the Minkowski space, in: Deformations of Mathematical Structures. Complex Analysis and Physical Applications, Selected papers from the Seminar on Deformations, Łódź-Lublin 1985/87, J. Ławrynowicz (ed.), Kluwer, Dordrecht 1989, 225-232.
  • [15] K. Kędzia, Hyperfine interactions versus intrinsic symmetries of many-electron systems in near-surface regions, Acta Phys. Superficiaerum 2 (1991), to appear.
  • [16] W. Królikowski, On correspondence between equations of motion for Dirac particle in curved and twisted space-times, ibid. 57 (1987), 143-153.
  • [17] W. Królikowski, Anisotropic complex structure on pseudo-Euclidean Hurwitz pairs, this volume, 225-240.
  • [18] G. L. Lamb, Jr., Elements of Soliton Theory, Wiley, New York 1980.
  • [19] J. Ławrynowicz (in cooperation with J. Krzyż), Quasiconformal Mappings in the Plane. Parametrical Methods, Lecture Notes in Math. 978, Springer, Berlin 1983.
  • [20] J. Ławrynowicz and J. Rembieliński, Hurwitz pairs equipped with complex structures, in: Seminar on Deformations, Proc. Łódź-Warsaw 1982/84. J. Ławrynowicz (ed.), Lecture Notes in Math. 1165, Springer, Berlin 1985, 184-195.
  • [21] J. Ławrynowicz and J. Rembieliński, Supercomplex vector spaces and spontaneous symmetry breaking, in: Seminari di Geometria 1984, Universitá di Bologna, Bologna 1985, 131-154.
  • [22] J. Ławrynowicz and J. Rembieliński, Pseudo-euclidean Hurwitz pairs and generalized Fueter equations, in: Clifford Algebras and Their Applications in Mathematical Physics, Proc. Canterbury 1985, J.S.R. Chisholm and A.K. Common (eds.), Reidel, Dordrecht 1986, 39-48.
  • [23] J. Ławrynowicz and J. Rembieliński, Complete classification for pseudo-euclidean Hurwitz pairs including the symmetry operations, Bull. Soc. Sci. Lettres Łódź 36 no. 29 (Série: Recherches sur les déformations, Volume '50 Years Anniversary of Scientific Work of Zygmunt Charzyński', 4 no. 39) (1986), 13 pp.
  • [24] J. Ławrynowicz and J. Rembieliński, Pseudo-euclidean Hurwitz pairs and the Kałuża-Klein theories, J. Phys. A. Math. Gen. 20 (1987), 5831-5848.
  • [25] J. Ławrynowicz and J. Rembieliński, On the composition of nondegenerate quadratic forms with an arbitrary index, Ann. Fac. Sci. Toulouse Math. (5) 10 (1989), 141-168 (due to a printing error in vol. 10, the whole article was reprinted in vol. 11 (1990), no. 1 of the same journal, pp. 141-168).
  • [26] M. Mulase, Complete integrability of the Kadomtsev-Petviashvili equation, Adv. in Math. 54 (1984), 57-66.
  • [27] M. Ohtsuka, Dirichlet Problem, Extremal Length and Prime Ends, Van Nostrand-Reinhold, New York 1970.
  • [28] M. Ohtsuka, Extremal length and precise functions in 3-space, Lecture Notes, Department of Math., Hiroshima Univ., Hiroshima 1973.
  • [29] R. Sasaki and I. Yamanaka, Virasoro algebra, vertex operators, quantum sine-Gordon and solvable quantum field theories, in: Conformal Field Theory and Solvable Lattice Models, Proc. Nagoya 1987, M. Jimbo, T. Miwa and A. Tsuchiya (eds.), Adv. Stud. in Pure Math. 16, Nagoya Univ. Press, Nagoya 1987.
  • [30] M. Sato, Soliton equations and Grassmann manifolds, lectures delivered at Nagoya Univ., 1982.
  • [31] J. H. Schwarz, Superstring theory, Phys. Rep. 89 (1982), 223-322.
  • [32] O. Suzuki, J. Ławrynowicz, J. Kalina And S. Kanemaki, A Geometric Approach To The Kadomtsev-petviashvili System I-ii, Proc. Inst. Nat. Sci. College Hum. Sci. Nihon Univ. 21 (1986), 11-34 And 24 (1989), to Appear.
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Bibliografia
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