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Geometric Structures Over Hypercomplex Algebras

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In this paper, we consider various algebras of hypercomplex numbers and geometric structures over them and discuss certain applications of these structures in theoretical physics.

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References

  1. D. Bambusi, “Clifford algebra description of non-Abelian gauge fields,” J. Geom. Phys., 7, No. 1, 1–12 (1990).

    Article  MathSciNet  Google Scholar 

  2. G. P. Barker and J. R. Urani, “Dirac general covariance and tetrads, I. Clifford and Lie bundles and torsion,” J. Math. Phys., 10, 2407–2410 (1983).

    MathSciNet  Google Scholar 

  3. M. Blau, “Connections on Clifford bundles and the Dirac operator,” Lett. Math. Phys., 13, No. 1, 83–92 (1987).

    Article  MathSciNet  Google Scholar 

  4. R. Brauer and H. Weyl, “Spinors in n dimensions,” Am. J. Math., 57, 425–449 (1935).

    Article  MathSciNet  Google Scholar 

  5. I. M. Burlakov and M. P. Burlakov, Introduction to Hypercomplex Analysis [in Russian], Lambert Academic Publishing (2017).

  6. I. M. Burlakov and M. P. Burlakov, Geometric Structures of Linear Algebras [in Russian], Lambert Academic Publishing (2017).

  7. M. P. Burlakov, V. V. Pokazeev, and L. E. Freidenzon, Clifford analysis, I. Clifford Δ-algebras [in Russian], deposited at the All-Russian Institute for Scientific and Technical Information, Moscow (1988), No. 1959-B88.

  8. M. P. Burlakov, “Clifford bundles and gauge fields,” Gravit. Teor. Otnosit., 23, 30–36 (1986).

    MathSciNet  Google Scholar 

  9. M. P. Burlakov, “Clifford connections on Riemannian spaces,” Izv. Vyssh. Ucheb. Zaved. Mat., No. 7, 3–7 (1990).

    MathSciNet  MATH  Google Scholar 

  10. M. P. Burlakov, “Clifford structures on manifolds,” in: Itogi Nauki Tekh. Sovr. Mat. Prilozh. Tematich. Obzory, 30, All-Russian Institute for Scientific and Technical Information, Moscow (2002), pp. 220–257.

  11. M. P. Burlakov, V. V. Pokazeev, and L. E. Freidenzon, Clifford analysis, II. Integral representation [in Russian], deposited at the All-Russian Institute for Scientific and Technical Information, Moscow (1988), No. 1960-B88.

  12. M. P. Burlakov, V. V. Pokazeev, and L. E. Freidenzon, Clifford analysis, II. Taylor series [in Russian], deposited at the All-Russian Institute for Scientific and Technical Information, Moscow (1988), No. 1961-B88.

  13. E. Cartan, La Théorie des Spineurs, Hermann, Paris (1938).

    MATH  Google Scholar 

  14. W. K. Clifford, “Applications of Grassmann’s extensive algebra,” Am. J. Math., 1, 350–358 (1878).

    Article  MathSciNet  Google Scholar 

  15. R. Delanghe, F. Brackx, and F. Sommen, Clifford Analysis, Pitnam, Boston–London–Melbourne (1982).

    MATH  Google Scholar 

  16. A. Dimakis and F. Muller-Hoissen, “Clifford calculus with applications to classical field theories,” Class. Quantum Grav., 8, No. 11, 2093–2132 (1991).

    Article  Google Scholar 

  17. P. A. M. Dirac, Creation of the Quantum Field Theory [Russian translation of selected papers], Nauka, Moscow (1990).

    Google Scholar 

  18. G. Dixon, “Fermionic Clifford algebras and supersymmetry,” in: Clifford Algebras and Their Applications in Mathematical Physics. Proc. NATO and SERC Workshop, Canterbury, Sept 15- 27, 1985, Dordrecht (1986), pp. 393–398.

  19. G. I. Garas’ko, Foundations of Finsler Geometry fot Physicists [in Russian], Moscow (2009).

  20. H. G. Grassmann, Gesammelte mathematische und physikalische Werke, Berlin (1894–1911).

  21. W. R. Hamilton, Lectures on Quaternions, Dublin (1853).

  22. A. P. Kotelnikov, Screw Calculus and Its Applications to Geometry and Mechanics [in Russian], Kazan’ (1895).

  23. A. Maia (Jr.), E. Recami, R. Valbur (Jr.), and A. F. Rosa Marcio, “Magnetic monopoles without string in the Kaller–Clifford algebra bundle: a geometrical interpretation,” J. Math. Phys., 31, No. 2, 502–505 (1990).

  24. W. Marcinek, “Clifford structures on real vector spaces and manifolds,” Repts. Math. Phys., 27, No. 3, 361–375 (1989).

    Article  MathSciNet  Google Scholar 

  25. I. P. Proskuryakov, Problems in Linear Algebra [in Russian], Nauka, Moscow (1967).

    Google Scholar 

  26. B. A. Rosenfeld, Non-Euclidean Geometries [in Russian], Gostkhizdat, Moscow (1954).

    Google Scholar 

  27. H. Rund, The Differential Geometry of Finsler Spaces, Springer-Verlag (1959).

  28. V. V. Vishnevsky, A. P. Shirokov, and V. V. Shurygin, Spaces over Algebras [in Russian], Kazan’ State Univ., Kazan’ (1985).

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Correspondence to I. M. Burlakov.

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Translated from Itogi Nauki i Tekhniki, Seriya Sovremennaya Matematika i Ee Prilozheniya. Tematicheskie Obzory, Vol. 145, Geometry and Mechanics, 2018.

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Burlakov, I.M., Burlakov, M.P. Geometric Structures Over Hypercomplex Algebras. J Math Sci 245, 538–552 (2020). https://doi.org/10.1007/s10958-020-04710-7

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