A theory of noncommutative determinants and characteristic functions of graphs

I. M. Gel'fand ; V. S. Retakh

Funkcionalʹnyj analiz i ego priloženiâ, Tome 26 (1992) no. 4, pp. 1-20.

Bibliographie

[1] Artin E., Geometric Algebra. Interscience, New York, 1957 | MR

[2] Artin M., Shelter W., Tate J., “Quantum deformations of $GL_n$”, Comm. on Pure and Appl. Math., 44 (1991), 879–895 | DOI | MR | Zbl

[3] Caeley A., “On certain results relating to quaternions”, Phil. Mag., 26 (1845), 141–145

[4] Capelli A., “Über die Zurtckfthrung der Caeley'shen Operationen $W$ und gewohnlisbe Polar operationen”, Math. Ann., 29 (1887), 331–338 | DOI | MR

[5] Berezin F.A., Introduction to Superanalysis, Dordrecht, Reidel, 1987 | MR | Zbl

[6] Bergman G.M, “Skew-fields of noncoramutative rational functions, after Amitsur”, Sem. Schtiizenberger–Lentin–Nivat 1969/70, 1970 | MR

[7] Bergman G.M., “Rational relations and rational identities in division rings. I, II”, J. Algebra, 43 (1976), 252–266 ; 267–297 | DOI | MR | Zbl | MR | Zbl

[8] Cohn P., Skew-Field Constructions, London Math. Soc. Lecture Note Series, 27, 1977 | MR | Zbl

[9] Cartier P., Foata D., Problemes combinatorics de commutation et rearrangements, Springer Lecture Notes, 85, 1969 | DOI | MR

[10] Bass H., Algebraic K-theory, W. A. Benjamin, New York, 1968 | MR | Zbl

[11] Diedonne J., “Les determinantes sur une corps non commutatif”, Bull. Soc. Math. France, 71 (1943), 27–45 | DOI | MR

[12] Diedonne J., La geometric des groupes classiques, Springer, Berlin, 1963 | MR

[13] Foata D., “A noncommutative version of the matrix inversion formula”, Adv. in Math., 31:3 (1979), 330–349 | DOI | MR | Zbl

[14] Foata D., “A combinatorial proof of Jacoby's identity”, Ann. Discrete Math., 6 (1980), 125–135 | DOI | MR | Zbl

[15] Gantmakher F.P., Teoriya matrits, Nauka, M., 1988 | MR | Zbl

[16] Gelfand I.M., Retakh V.S., “Determinanty matrits nad nekommutativnymi koltsami”, Funktsion. analiz i ego pril., 25:2 (1991), 13–25 | MR

[17] Heyting A., “Die Theorie der linearen Gleichungen in einer Zahlenspezies mit nichtkommutativer Multiplikation”, Math. Ann., 98 (1927), 465–490 | DOI | MR | Zbl

[18] Howe V., Umeda T., “The Capelli identity, the double commutant theorem, and the multiciplicities-free actions”, Math. Ann., 290 (1991), 565–619 | DOI | MR | Zbl

[19] Joly J., Supplement to Oevres of Hamilton, 1900

[20] Kulish P.P., Sklyanin E.K., “Quantum spectral transform method. Recent developments”, Lecture Notes in Phys., 151, 1982, 61–119 | DOI | MR | Zbl

[21] Kostant B., Sahi S., “The Capelli identity, tube domains and the generalized Laplace transform”, Adv. Math., 87 (1991), 71–92 | DOI | MR | Zbl

[22] Manin Yu.I., “Multiparameter quantum deformations of the general linear suppergroup”, Comm. Math. Phys., 123 (1989), 163–175 | DOI | MR | Zbl

[23] Okamoto K., “Backlund Transformations of Classical Orthogonal Polynomials”, Algebraic Analysis, vol. 1, eds. M. Kashiwara and T. Kawai, 1988, 647–657 | DOI | MR

[24] Ore O., “Linear equations in non-commutative rings”, Ann. Math., 32 (1931), 463–477 | DOI | MR

[25] Richardson A.R., “Hypercomplex determinants”, Messenger of Math., 55 (1926), 145–152 | Zbl

[26] Richardson A.R., “Simultaneous linear equations over a division algebra”, London Math. Soc., 28 (1928), 395–420 | DOI | MR | Zbl

[27] Sato M., Kashiwara H., “The determinant of matrices of pseudodifferential operators”, Proc. Japan Acad. Sci., 51:1 (1975), 17–19 | DOI | MR | Zbl

[28] Wedderburn J.H.M., “On continued fractions in non-commutative quantities”, Ann. Math., 15:14 (1913), 101–105 | DOI | MR

[29] Weyl H., Classical Groups, their invariants and representations, Princeton Univ. Press, 1946 | MR | Zbl

[30] Williams R., Preprint in determinants in symbolic dynamics, University of Texas at Austin, 1991