Quasideterminants.(English)Zbl 1079.15007
The authors present a survey on quasideterminants for matrices over a division ring. In the introduction they outline the many attempts of defining a determinant for matrices with non-commutative entries in the past 160 years.
In the main part of the paper the authors (1) give a definition of quasideterminants and describe their main properties, (2) discuss quasideterminants of quaternionic matrices, (3) give a general definition of determinants of non-commutative matrices based on the notion of quasideterminants and show how to obtain some well-known non-commutative determinants as specializations, (4) introduce non-commutative Plücker and flag coordinates for rectangular matrices over a division ring, (5) discuss how to factorize a non-commutative univariate polynomial into products of linear polynomials (non-commutative Bézout theorem) and how to express the coefficients of a non-commutative univariate polynomial in terms of its roots (noncommutative Viète theorem), (6) give a theory of non-commutative symmetric functions, (7) construct universal quadratic algebras associated with so-called pseudo-roots of non-commutative polynomials and non-commutative differential polynomials, (8) present another approach to the theory of non-commutative determinants by using cyclic vectors and relate this to the results of (3) and (5), (9) give some applications to non-commutative continued fractions, characteristic functions of graphs, factorizations of differential operators and non-commutative variation of constants, non-commutative integrable systems, and non-commutative orthogonal polynomials.
The paper concludes with an extensive bibliography on the subject.
In the main part of the paper the authors (1) give a definition of quasideterminants and describe their main properties, (2) discuss quasideterminants of quaternionic matrices, (3) give a general definition of determinants of non-commutative matrices based on the notion of quasideterminants and show how to obtain some well-known non-commutative determinants as specializations, (4) introduce non-commutative Plücker and flag coordinates for rectangular matrices over a division ring, (5) discuss how to factorize a non-commutative univariate polynomial into products of linear polynomials (non-commutative Bézout theorem) and how to express the coefficients of a non-commutative univariate polynomial in terms of its roots (noncommutative Viète theorem), (6) give a theory of non-commutative symmetric functions, (7) construct universal quadratic algebras associated with so-called pseudo-roots of non-commutative polynomials and non-commutative differential polynomials, (8) present another approach to the theory of non-commutative determinants by using cyclic vectors and relate this to the results of (3) and (5), (9) give some applications to non-commutative continued fractions, characteristic functions of graphs, factorizations of differential operators and non-commutative variation of constants, non-commutative integrable systems, and non-commutative orthogonal polynomials.
The paper concludes with an extensive bibliography on the subject.
Reviewer: Jörg Feldvoss (Mobile)
MSC:
| 15A15 | Determinants, permanents, traces, other special matrix functions |
| 05E05 | Symmetric functions and generalizations |
| 16W30 | Hopf algebras (associative rings and algebras) (MSC2000) |
| 15B33 | Matrices over special rings (quaternions, finite fields, etc.) |
Keywords:
Quasideterminants;non-commutative determinants;non-commutative algebra;non-commutative geometry;non-commutative symmetric functions;non-commutative continued fractions;non-commutative integrable systems;division ring;quaternionic matricesCite
References:
| [1] | S. Alesker, Non-commutative linear algebra and plurisubharmonic functions of quaternionic variables, arXiv: math.CV/010429 (2002).; S. Alesker, Non-commutative linear algebra and plurisubharmonic functions of quaternionic variables, arXiv: math.CV/010429 (2002). |
| [2] | Artin, E., Geometric Algebra (1988), Wiley: Wiley New York ·Zbl 0642.51001 |
| [3] | Aslaksen, H., Quaternionic determinants, Math. Intell., 18, 3, 57-65 (1996) ·Zbl 0881.15007 |
| [4] | F.A. Berezin, Introduction to Algebra and Analysis with Anticommuting Variables, Izd. Moskov. Univ. Moscow 1983 (in Russian).; F.A. Berezin, Introduction to Algebra and Analysis with Anticommuting Variables, Izd. Moskov. Univ. Moscow 1983 (in Russian). ·Zbl 0527.15020 |
| [5] | Birman, J.; Williams, R., Knotted periodic orbits ILorenz knots, Topology, 22, 47-62 (1983) ·Zbl 0507.58038 |
| [6] | Cartier, P.; Foata, D., Problemes combinatoires de commutation et rearrangements. Problemes combinatoires de commutation et rearrangements, Lecture Notes in Mathematics, Vol. 85 (1969), Springer: Springer Berlin, Hiedelberg ·Zbl 0186.30101 |
| [7] | Cayley, A., On certain results relating to quaternions, Philos. Mag., 26, 141-145 (1845) |
| [8] | Cohn, P. M., Skew-field Constructions. Skew-field Constructions, London Mathematical Society Lecture Notes, Vol. 27 (1977), Cambridge University Press: Cambridge University Press Cambridge ·Zbl 0355.16009 |
| [9] | Connes, A.; Schwarz, A., Matrix Viète theorem revisited, Lett. Math. Phys., 39, 349-353 (1997) ·Zbl 0874.15010 |
| [10] | Dieudonne, J., Les determinantes sur un corps non-commutatif, Bull. Soc. Math. France, 71, 27-45 (1943) ·Zbl 0028.33904 |
| [11] | P.K. Draxl, London Mathematical Society Lecture Note Series, Vol. 81, Cambridge University Press, Cambridge, 1983.; P.K. Draxl, London Mathematical Society Lecture Note Series, Vol. 81, Cambridge University Press, Cambridge, 1983. ·Zbl 0498.16015 |
| [12] | V.G. Drinfeld, V.V. Sokolov, Lie algebras and equations of Korteweg-de Vries type, Journal of Current Problems in Mathematics, Itogi Nauki i Tekniki, Akad. Nauk. SSSR, Vol. 24, VINITI, Moscow, 1984, pp. 81-180 (in Russian).; V.G. Drinfeld, V.V. Sokolov, Lie algebras and equations of Korteweg-de Vries type, Journal of Current Problems in Mathematics, Itogi Nauki i Tekniki, Akad. Nauk. SSSR, Vol. 24, VINITI, Moscow, 1984, pp. 81-180 (in Russian). ·Zbl 0558.58027 |
| [13] | Dyson, F. J., Correlations between eigenvalues of random matrices, Comm. Math. Phys., 19, 235-250 (1970) ·Zbl 0221.62019 |
| [14] | Dyson, F. J., Quaternion determinants, Helv. Phys. Acta, 45, 289-302 (1972) |
| [15] | Etingof, P.; Gelfand, I.; Retakh, V., Factorization of differential operators, quasideterminants, and nonabelian Toda field equations, Math. Res. Lett., 4, 2-3, 413-425 (1997) ·Zbl 0959.37054 |
| [16] | Etingof, P.; Gelfand, I.; Retakh, V., Nonabelian integrable systems, quasideterminants, and Marchenko lemma, Math. Res. Lett., 5, 1-2, 1-12 (1998) ·Zbl 0948.37055 |
| [17] | Etingof, P.; Retakh, V., Quantum determinants and quasideterminants, Asian J. Math., 3, 2, 345-352 (1999) ·Zbl 0985.17013 |
| [18] | Foata, D., A noncommutative version of the matrix inversion formula, Adv. in Math., 31, 330-349 (1979) ·Zbl 0418.15004 |
| [19] | Frobenius, G., Über lineare Substituenen und bilinear Formen, J. Reine Angew. Math., 84, 1-63 (1877) ·JFM 09.0085.02 |
| [20] | Friedberg, S. H.; Insel, A. J.; Spence, L. E., Linear Algebra (1997), Prentice-Hall: Prentice-Hall Upper Saddle River, NJ ·Zbl 0864.15001 |
| [21] | Gantmacher, F. R., The Theory of Matrices (1959), Chelsea: Chelsea New York ·Zbl 0085.01001 |
| [22] | Gelfand, I.; Gelfand, S.; Retakh, V., Noncommutative algebras associated to complexes and graphs, Selecta Math. (N.S.), 7, 525-531 (2001) ·Zbl 0992.16026 |
| [23] | Gelfand, I.; Gelfand, S.; Retakh, V.; Serconek, S.; Wilson, R., Hilbert series of quadratic algebras associated with decompositions of noncommutative polynomials, J. Algebra, 254, 279-299 (2002) ·Zbl 1015.16034 |
| [24] | Gelfand, I.; Krob, D.; Lascoux, A.; Leclerc, B.; Retakh, V.; Thibon, J.-Y., Noncommutative symmetric functions, Adv. in Math., 112, 2, 218-348 (1995) ·Zbl 0831.05063 |
| [25] | Gelfand, I.; Retakh, V., Determinants of matrices over noncommutative rings, Funct. Anal. Appl., 25, 2, 91-102 (1991) ·Zbl 0748.15005 |
| [26] | Gelfand, I.; Retakh, V., A theory of noncommutative determinants and characteristic functions of graphs, Funct. Anal. Appl., 26, 4, 1-20 (1992) ·Zbl 0799.15003 |
| [27] | Gelfand, I.; Retakh, V., A theory of noncommutative determinants and characteristic functions of graphs. I, Publ. LACIM, UQAM, Montreal, 14, 1-26 (1993) |
| [28] | I. Gelfand, V. Retakh, Noncommutative Vieta theorem and symmetric functions, Gelfand Mathematical Seminars 1993-95, Birkhäuser Boston, 1996, pp. 93-100; I. Gelfand, V. Retakh, Noncommutative Vieta theorem and symmetric functions, Gelfand Mathematical Seminars 1993-95, Birkhäuser Boston, 1996, pp. 93-100 ·Zbl 0865.05074 |
| [29] | Gelfand, I.; Retakh, V., Quasideterminants, I, Selecta Math. (N.S.), 3, 4, 517-546 (1997) ·Zbl 0919.16011 |
| [30] | Gelfand, I.; Retakh, V.; Wilson, R., Quadratic-linear algebras associated with decompositions of noncommutative polynomials and differential polynomials, Selecta Math. (N.S.), 7, 493-523 (2001) ·Zbl 0992.16025 |
| [31] | I. Gelfand, V. Retakh, R. Wilson, Quaternionic quasideterminants and determinants, Lie groups and symmetric spaces, American Mathematical Society Translation Series 2, Vol. 210, American Mathematical Society, Providence, RI, 2003, pp. 111-123.; I. Gelfand, V. Retakh, R. Wilson, Quaternionic quasideterminants and determinants, Lie groups and symmetric spaces, American Mathematical Society Translation Series 2, Vol. 210, American Mathematical Society, Providence, RI, 2003, pp. 111-123. ·Zbl 1037.15006 |
| [32] | A. Givental, Stationary Phase Integrals, Quantum Toda Lattices, Flag Manifolds and the Mirror Conjecture. Topics in Singularity Theory, American Mathematical Society Translation Series 2, Vol 180, American Mathematical Society, Providence, RI, 1997, pp. 103-115.; A. Givental, Stationary Phase Integrals, Quantum Toda Lattices, Flag Manifolds and the Mirror Conjecture. Topics in Singularity Theory, American Mathematical Society Translation Series 2, Vol 180, American Mathematical Society, Providence, RI, 1997, pp. 103-115. ·Zbl 0895.32006 |
| [33] | Heyting, A., Die Theorie der linearen Gleichungen in einer Zahlenspezies mit nichtkommutativer Multiplication, Math. Ann., 98, 465-490 (1927) ·JFM 53.0120.01 |
| [34] | R. Howe, Perspectives in invariant theory: Schur duality, multiplicity-free actions and beyond Israel Mathematical Conference Proceedings, Vol. 8, Bar-Ilan University, Ramat Gan, 1995, pp. 1-182.; R. Howe, Perspectives in invariant theory: Schur duality, multiplicity-free actions and beyond Israel Mathematical Conference Proceedings, Vol. 8, Bar-Ilan University, Ramat Gan, 1995, pp. 1-182. ·Zbl 0844.20027 |
| [35] | Jacobson, N., Structure and Representations of Jordan Algebras. Structure and Representations of Jordan Algebras, Colloquium Publications, Vol. 39 (1968), American Mathematical Society: American Mathematical Society Providence, RI ·Zbl 0218.17010 |
| [36] | N. Jacobson, Lectures in Abstract Algebra, Vol. II, Springer, New York, Berlin, 1975.; N. Jacobson, Lectures in Abstract Algebra, Vol. II, Springer, New York, Berlin, 1975. ·Zbl 0314.15001 |
| [37] | Joly, C., Supplement to Oevres of Hamilton, (Hamilton, W. R., Elements of Quaternions (1969), Chelsea: Chelsea New York) |
| [38] | Kapranov, M., Noncommutative geometry based on commutator expansions, J. Reine Angew. Math., 505, 73-118 (1998) ·Zbl 0918.14001 |
| [39] | Kirillov, A., Introduction to family algebras, Moscow Math. J., 1, 1, 49-63 (2001) ·Zbl 0981.15013 |
| [40] | M. Kontsevich, A. Rosenberg, Noncommutative smooth spaces The Gelfand Mathematical Seminars 1996-1999, Birkhäuser Boston, 2000, pp. 85-108.; M. Kontsevich, A. Rosenberg, Noncommutative smooth spaces The Gelfand Mathematical Seminars 1996-1999, Birkhäuser Boston, 2000, pp. 85-108. ·Zbl 1003.14001 |
| [41] | Kostant, B., The solution to a generalized Toda lattice and representation theory, Adv. in Math., 34, 195-338 (1979) ·Zbl 0433.22008 |
| [42] | Krob, D.; Leclerc, B., Minor identities for quasi-determinants and quantum determinants, Comm. Math. Phys., 169, 1, 1-23 (1995) ·Zbl 0829.15024 |
| [43] | Krob, D.; Leclerc, B.; Thibon, J.-Y., Noncommutative symmetric functions. II. Transformations of alphabets, Internat. J. Algebra Comput., 7, 181-264 (1997) ·Zbl 0907.05055 |
| [44] | P.P. Kulish, E.K. Sklyanin, Quantum Spectral Transform Method. Recent Developments, Lecture Notes in Physics, Vol. 151, Springer, Berlin, Heidelberg, New York, 1982, pp. 61-119.; P.P. Kulish, E.K. Sklyanin, Quantum Spectral Transform Method. Recent Developments, Lecture Notes in Physics, Vol. 151, Springer, Berlin, Heidelberg, New York, 1982, pp. 61-119. ·Zbl 0734.35071 |
| [45] | Lam, T. Y., Representations of finite groupsa hundred years, Part I, Notices Amer. Math. Soc., 45, 4, 361-372 (1998) ·Zbl 0924.01008 |
| [46] | Leclerc, B.; Scharf, T.; Thibon, J.-Y., Noncommutative cyclic characters of symmetric groups, J. Combin. Theory A, 75, 55-69 (1996) ·Zbl 0857.20005 |
| [47] | Leites, D. A., A certain analog of the determinant, Russian Math. Surv., 35, 3, 156 (1975), (in Russian) ·Zbl 0335.15011 |
| [48] | Löfwall, On the subalgebra generated by the one-dimensional elements in the Yoneda Ext-algebra, Lecture Notes Math. 1183 (1986) 291-338.; Löfwall, On the subalgebra generated by the one-dimensional elements in the Yoneda Ext-algebra, Lecture Notes Math. 1183 (1986) 291-338. ·Zbl 0595.16020 |
| [49] | Macdonald, I., Symmetric Functions and Hall Polynomials (1995), The Clarendon Press, Oxford University Press: The Clarendon Press, Oxford University Press New York ·Zbl 0824.05059 |
| [50] | MacMacon, P. A., Combinatorial Analysis (1915), Cambridge University Press: Cambridge University Press Cambridge ·JFM 45.1271.01 |
| [51] | Manin, Y., Multiparameter quantum deformations of the linear supergroup, Comm. Math. Phys., 113, 163-175 (1989) ·Zbl 0673.16004 |
| [52] | V.B. Matveev, Darboux transformations, covariance theorems and integrable systems, in: M. Semenov-Tian-Shansky (Ed.), L.D. Faddeev’s Seminar on Mathematical Physics, American Mathematical Society Translation Series 2, Vol. 201, American Mathematical Society, Providence, RI, 2000, pp. 179-209.; V.B. Matveev, Darboux transformations, covariance theorems and integrable systems, in: M. Semenov-Tian-Shansky (Ed.), L.D. Faddeev’s Seminar on Mathematical Physics, American Mathematical Society Translation Series 2, Vol. 201, American Mathematical Society, Providence, RI, 2000, pp. 179-209. ·Zbl 0963.37066 |
| [53] | Molev, A., Noncommutative symmetric functions and Laplace operators for classical Lie algebras, Lett. Math. Phys., 35, 2, 135-143 (1995) ·Zbl 0843.17003 |
| [54] | Molev, A., Gelfand-Tsetlin bases for representations of Yangians, Lett. Math. Phys., 30, 2, 53-60 (1994) ·Zbl 0787.17014 |
| [55] | Molev, A.; Retakh, V., Quasideterminants and Casimir elements for the general Lie superalgebras, Internat. Math. Res. Notes, 13, 611-619 (2004) ·Zbl 1079.17006 |
| [56] | Moore, E. H., On the determinant of an Hermitian matrix of quaternionic elements, Bull. Amer. Math. Soc., 28, 161-162 (1922) ·JFM 48.0128.07 |
| [57] | Moore, E. H.; Barnard, R. W., General Analysis Part. I (1935), The American Philosophical Society: The American Philosophical Society Philadelphia ·Zbl 0013.11605 |
| [58] | Okamoto, K., Bäcklund transformations of classical orthogonal polynomials, (Kashiwara, M.; Kawai, T., Algebraic Analysis II (1988), Academic Press: Academic Press Boston, MA), 647-657 ·Zbl 0682.33008 |
| [59] | Ore, O., Liner equations in noncommutative rings, Ann. Math., 32, 463-477 (1931) ·JFM 57.0166.01 |
| [60] | B.L. Osofsky, Quasideterminants and roots of polynomials over division rings, Proceedings of the International Lisboa Conference on Algebras, Modules and Rings, 2003 (to appear).; B.L. Osofsky, Quasideterminants and roots of polynomials over division rings, Proceedings of the International Lisboa Conference on Algebras, Modules and Rings, 2003 (to appear). ·Zbl 1110.15009 |
| [61] | I. Pak, A. Postnikov, V. Retakh, Noncommutative Lagrange inversion, 1995 (preprint).; I. Pak, A. Postnikov, V. Retakh, Noncommutative Lagrange inversion, 1995 (preprint). |
| [62] | A. Polischuk, Triple Massey products on curves, Fay’s trisecant identity and tangents to the canonical embedding, math.AG/0109007 (2001).; A. Polischuk, Triple Massey products on curves, Fay’s trisecant identity and tangents to the canonical embedding, math.AG/0109007 (2001). |
| [63] | Razumov, A.; Saveliev, M., Maximally nonabelian Toda systems, Nucl. Phys. B, 494, 3, 657-686 (1997) ·Zbl 0996.37508 |
| [64] | V. Retakh, C. Reutenauer, A. Vaintrob, Noncommutative rational functions and Farber’s invariants of boundary links, in: A. Astashkevich, S. Tabachnikov (Eds.), Differential Topology, Infinite-dimensional Lie Algebras and Applications, American Mathematical Society Translation Series 2, Vol. 194, American Mathematical Society, Providence, RI, 1999, pp. 237-246.; V. Retakh, C. Reutenauer, A. Vaintrob, Noncommutative rational functions and Farber’s invariants of boundary links, in: A. Astashkevich, S. Tabachnikov (Eds.), Differential Topology, Infinite-dimensional Lie Algebras and Applications, American Mathematical Society Translation Series 2, Vol. 194, American Mathematical Society, Providence, RI, 1999, pp. 237-246. ·Zbl 0971.68103 |
| [65] | V. Retakh, V. Shander, Schwarz derivative for noncommutative differential algebras, in: D. Fuchs (Ed.), Unconvential Lie Algebras Advances in Soviet Mathematics, Vol. 17, American Mathematical Society, Providence, RI, 1993, pp. 130-154.; V. Retakh, V. Shander, Schwarz derivative for noncommutative differential algebras, in: D. Fuchs (Ed.), Unconvential Lie Algebras Advances in Soviet Mathematics, Vol. 17, American Mathematical Society, Providence, RI, 1993, pp. 130-154. ·Zbl 0841.17003 |
| [66] | Reutenauer, C., Inversion height in free fields, Selecta Math. (N.S.), 2, 1, 93-109 (1996) ·Zbl 0857.16021 |
| [67] | Richardson, A. R., Hypercomplex determinants, Messenger Math., 55, 145-152 (1926) ·JFM 52.0134.01 |
| [68] | Richardson, A. R., Simultaneous linear equations over a division algebra, Proc. London Math. Soc., 28, 395-420 (1928) ·JFM 54.0161.06 |
| [69] | Schork, M., The bc-system of higher rank revisited, J. Phys. A: Math. Gen., 35, 2627-2637 (2001) ·Zbl 1026.81049 |
| [70] | Schur, I., Über die charakteristischen Wurzeln einer linearen Substitution mit einer Anwendung auf die Theorie der Integralgleichungen, Math. Ann., 66, 488-510 (1909) ·JFM 40.0396.03 |
| [71] | S. Serconek, R.L. Wilson, Quadratic algebras associated with decompositions of noncommutative polynomials are Koszul algebras, in preparation.; S. Serconek, R.L. Wilson, Quadratic algebras associated with decompositions of noncommutative polynomials are Koszul algebras, in preparation. ·Zbl 1062.16036 |
| [72] | Stafford, J. T.; van den Bergh, M., Noncommutative curves and noncommutative surfaces, Bull. Amer. Math. Soc., 38, 2, 171-216 (2001) ·Zbl 1042.16016 |
| [73] | Study, E., Zur Theorie der lineare Gleichungen, Acta Math., 42, 1-61 (1920) ·JFM 46.0144.06 |
| [74] | Sylvester, J. J., Sur une classe nouvelle d’equations differéntielles et d’equations aux differences finies d’une forme intégrable, Compt. Rend. Acad. Sci., 54, 129-132170 (1862) |
| [75] | Thibon, J.-Y., Lectures on noncommutative symmetric functions, MSJ Mem., 11, 39-94 (2001) ·Zbl 0990.05136 |
| [76] | Van Praag, P., Sur le norme reduite du determinant de Dieudonne des matrices quaterniennes, J. Algebra, 136, 265-274 (1991) ·Zbl 0724.15013 |
| [77] | Wedderburn, J. H.M., On continued fractions in noncommutative quantities, Ann. Math., 15, 101-105 (1913) ·JFM 45.0341.01 |
| [78] | Weyl, H., Classical Groups, their Invariants and Representations (1946), Princeton University Press: Princeton University Press Princeton, NJ ·Zbl 1024.20502 |
| [79] | Wilson, R. L., Invariant polynomials in the free skew field, Selecta Math., 7, 565-586 (2001) ·Zbl 0992.16016 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
