Evaluations of Noncommutative Polynomials on Algebras: Methods and Problems, and the L'vov–Kaplansky Conjecture

Alexei Kanel-Belov ; Sergey Malev ; Louis Rowen ; Roman Yavich

Symmetry, integrability and geometry: methods and applications, Tome 16 (2020)

Résumé

Let $p$ be a polynomial in several non-commuting variables with coefficients in a field $K$ of arbitrary characteristic. It has been conjectured that for any $n$, for $p$ multilinear, the image of $p$ evaluated on the set $M_n(K)$ of $n$ by $n$ matrices is either zero, or the set of scalar matrices, or the set ${\rm sl}_n(K)$ of matrices of trace 0, or all of $M_n(K)$. This expository paper describes research on this problem and related areas. We discuss the solution of this conjecture for $n=2$ in Section 2, some decisive results for $n=3$ in Section 3, and partial information for $n\geq 3$ in Section 4, also for non-multilinear polynomials. In addition we consider the case of $K$ not algebraically closed, and polynomials evaluated on other finite dimensional simple algebras (in particular the algebra of the quaternions). This review recollects results and technical material of our previous papers, as well as new results of other researches, and applies them in a new context. This article also explains the role of the Deligne trick, which is related to some nonassociative cases in new situations, underlying our earlier, more straightforward approach. We pose some problems for future generalizations and point out possible generalizations in the present state of art, and in the other hand providing counterexamples showing the boundaries of generalizations.

Détail

Keywords: L'vov–Kaplansky conjecture, noncommutative polynomials, multilinear polynomial evaluations, power central polynomials, the Deligne trick, PI algebras.

Bibliographie

[1] Albert A. A., Muckenhoupt B., “On matrices of trace zeros”, Michigan Math. J., 4 (1957), 1–3 | DOI | MR | Zbl

[2] Almutairi N. B., On multilinear polynomials evaluated on quaternion algebra, Master Thesis, Kent State University, 2016

[3] Amitsur A. S., Levitzki J., “Minimal identities for algebras”, Proc. Amer. Math. Soc., 1 (1950), 449–463 | DOI | MR | Zbl

[4] Anzis B. E., Emrich Z. M., Valiveti K. G., “On the images of Lie polynomials evaluated on Lie algebras”, Linear Algebra Appl., 469 (2015), 51–75 | DOI | MR | Zbl

[5] Bahturin Yu.A., Identical relations in Lie algebras, VNU Science Press, Utrecht, 1987 | MR | Zbl

[6] Bandman T., Garion S., Grunewald F., “On the surjectivity of Engel words on ${\rm PSL}(2,q)$”, Groups Geom. Dyn., 6 (2012), 409–439, arXiv: 1008.1397 | DOI | MR | Zbl

[7] Bandman T., Garion S., Kunyavskii B., “Equations in simple matrix groups: algebra, geometry, arithmetic, dynamics”, Cent. Eur. J. Math., 12 (2014), 175–211, arXiv: 1302.4667 | DOI | MR | Zbl

[8] Bandman T., Gordeev N., Kunyavskii B., Plotkin E., “Equations in simple Lie algebras”, J. Algebra, 355 (2012), 67–79, arXiv: 1012.4106 | DOI | MR | Zbl

[9] Bandman T., Kunyavskii B., “Criteria for equidistribution of solutions of word equations on ${\rm SL}(2)$”, J. Algebra, 382 (2013), 282–302, arXiv: 1201.5260 | DOI | MR | Zbl

[10] Bandman T., Zarhin Y. G., “Surjectivity of certain word maps on ${\rm PSL}(2,\mathbb{C})$ and ${\rm SL}(2,\mathbb{C})$”, Eur. J. Math., 2 (2016), 614–643, arXiv: 1407.3447 | DOI | MR | Zbl

[11] Belov A.Ya., “Counterexamples to the Specht problem”, Sb. Math., 191 (2000), 329–340 | DOI | MR | Zbl

[12] Belov A.Ya., Borisenko V. V., Latyshev V. N., “Monomial algebras”, J. Math. Sci., 87 (1997), 3463–3575 | DOI | MR | Zbl

[13] Bokut L. A., “Unsolvability of the equality problem and subalgebras of finitely presented Lie algebras”, Math. USSR-Izv., 6 (1972), 1153–1199 | DOI | MR

[14] Bokut L. A., Chen Y., “Gröbner–Shirshov bases and their calculation”, Bull. Math. Sci., 4 (2014), 325–395, arXiv: 1303.5366 | DOI | MR | Zbl

[15] Bokut L. A., Kukin G. P., “Undecidable algorithmic problems for semigroups, groups and rings”, J. Soviet Math., 45 (1989), 871–911 | DOI | MR | Zbl

[16] Borel A., “On free subgroups of semisimple groups”, Enseign. Math., 29 (1983), 151–164 | MR | Zbl

[17] Brešar M., Commutators and images of noncommutative polynomials, arXiv: 2001.10392

[18] Brešar M., Klep I., “Values of noncommutative polynomials, Lie skew-ideals and tracial Nullstellensätze”, Math. Res. Lett., 16 (2009), 605–626, arXiv: 0810.1774 | DOI | MR | Zbl

[19] Brešar M., Procesi C., Špenko Š., “Quasi-identities on matrices and the Cayley–Hamilton polynomial”, Adv. Math., 280 (2015), 439–471, arXiv: 1212.4597 | DOI | MR | Zbl

[20] Burness T. C., Liebeck M. W., Shalev A., “The length and depth of algebraic groups”, Math. Z., 291 (2019), 741–760, arXiv: 1712.08214 | DOI | MR | Zbl

[21] Burness T. C., Liebeck M. W., Shalev A., “The length and depth of compact Lie groups”, Math. Z., 294 (2020), 1457–1476, arXiv: 1805.09893 | DOI | MR | Zbl

[22] Buzinski D., Winstanley R., “On multilinear polynomials in four variables evaluated on matrices”, Linear Algebra Appl., 439 (2013), 2712–2719, arXiv: 1511.06331 | DOI | MR | Zbl

[23] Chuang C.-L., “On ranges of polynomials in finite matrix rings”, Proc. Amer. Math. Soc., 110 (1990), 293–302 | DOI | MR | Zbl

[24] Cohn P. M., “The range of derivations on a skew field and the equation $ax-xb=c$”, J. Indian Math. Soc. (N.S.), 37 (1973), 61–69 | MR

[25] Cox D., Little J., O'Shea D., Ideals, varieties, and algorithms. An introduction to computational algebraic geometry and commutative algebra, Undergraduate Texts in Mathematics, 3rd ed., Springer, New York, 2007 | DOI | MR | Zbl

[26] De Concini C., Papi P., Procesi C., “The adjoint representation inside the exterior algebra of a simple Lie algebra”, Adv. Math., 280 (2015), 21–46, arXiv: 1311.4338 | DOI | MR | Zbl

[27] De Concini C., Procesi C., The invariant theory of matrices, University Lecture Series, 69, Amer. Math. Soc., Providence, RI, 2017 | DOI | MR | Zbl

[28] Deligne P., Sullivan D., “Division algebras and the Hausdorff–Banach–Tarski paradox”, Enseign. Math., 29 (1983), 145–150 | MR | Zbl

[29] Dniester Notebook: Unsolved problems in the theory of rings and modules, 4th ed., Mathematics Institute, Russian Academy of Sciences Siberian Branch, Novosibirsk, 1993 | Zbl

[30] Donkin S., “Invariants of several matrices”, Invent. Math., 110 (1992), 389–401 | DOI | MR | Zbl

[31] Drensky V., Free algebras and PI-algebras, Graduate course in algebra, Springer-Verlag Singapore, Singapore, 2000 | MR | Zbl

[32] Drensky V., Formanek E., Polynomial identity rings, Advanced Courses in Mathematics. CRM Barcelona, Birkhäuser Verlag, Basel, 2004 | DOI | MR | Zbl

[33] Drensky V., Kasparian A., “A new central polynomial for $3\times 3$ matrices”, Comm. Algebra, 13 (1985), 745–752 | DOI | MR | Zbl

[34] Drensky V., Piacentini Cattaneo G. M., “A central polynomial of low degree for $4\times 4$ matrices”, J. Algebra, 168 (1994), 469–478 | DOI | MR | Zbl

[35] Drensky V., Rashkova T. G., “Weak polynomial identities for the matrix algebras”, Comm. Algebra, 21 (1993), 3779–3795 | DOI | MR | Zbl

[36] Dykema K. J., Klep I., “Instances of the Kaplansky–Lvov multilinear conjecture for polynomials of degree three”, Linear Algebra Appl., 508 (2016), 272–288, arXiv: 1508.01238 | DOI | MR | Zbl

[37] Egorchenkova E., Gordeev N., “Products of three word maps on simple algebraic groups”, Arch. Math. (Basel), 112 (2019), 113–122 | DOI | MR | Zbl

[38] Ellers E. W., Gordeev N., “On the conjectures of J. Thompson and O. Ore”, Trans. Amer. Math. Soc., 350 (1998), 3657–3671 | DOI | MR | Zbl

[39] Fagundes P. S., “The images of multilinear polynomials on strictly upper triangular matrices”, Linear Algebra Appl., 563 (2019), 287–301, arXiv: 1807.09136 | DOI | MR | Zbl

[40] Fagundes P. S., de Mello T. C., “Images of multilinear polynomials of degree up to four on upper triangular matrices”, Oper. Matrices, 13 (2019), 283–292, arXiv: 1807.09421 | DOI | MR | Zbl

[41] Formanek E., “Central polynomials for matrix rings”, J. Algebra, 23 (1972), 129–132 | DOI | MR | Zbl

[42] Formanek E., The polynomial identities and invariants of $n\times n$ matrices, CBMS Regional Conference Series in Mathematics, 78, Amer. Math. Soc., Providence, RI, 1991 | DOI | MR | Zbl

[43] Giambruno A., Zaicev M., Polynomial identities and asymptotic methods, Mathematical Surveys and Monographs, 122, Amer. Math. Soc., Providence, RI, 2005 | DOI | MR | Zbl

[44] Gordeev N., “On Engel words on simple algebraic groups”, J. Algebra, 425 (2015), 215–244 | DOI | MR | Zbl

[45] Gordeev N., Kunyavskii B., Plotkin E., “Geometry of word equations in simple algebraic groups over special fields”, Russian Math. Surveys, 73 (2018), 753–796, arXiv: 1808.02303 | DOI | MR | Zbl

[46] Gordeev N., Kunyavskii B., Plotkin E., “Word maps on perfect algebraic groups”, Internat. J. Algebra Comput., 28 (2018), 1487–1515, arXiv: 1801.00381 | DOI | MR | Zbl

[47] Gordeev N., Kunyavskii B., Plotkin E., “Word maps, word maps with constants and representation varieties of one-relator groups”, J. Algebra, 500 (2018), 390–424, arXiv: 1801.00379 | DOI | MR | Zbl

[48] Gordienko A. S., Lecture notes on polynomial identities, Preprint, 2014 | MR

[49] Guterman A. E., Kuzma B., “Maps preserving zeros of matrix polynomials”, Dokl. Math., 80 (2009), 508–510 | DOI | MR | Zbl

[50] Halpin P., “Central and weak identities for matrices”, Comm. Algebra, 11 (1983), 2237–2248 | DOI | MR | Zbl

[51] Helling H., “Eine Kennzeichnung von Charakteren auf Gruppen und assoziativen Algebren”, Comm. Algebra, 1 (1974), 491–501 | DOI | MR | Zbl

[52] Herstein I. N., “On the Lie structure of an associative ring”, J. Algebra, 14 (1970), 561–571 | DOI | MR | Zbl

[53] Hui C. Y., Larsen M., Shalev A., “The Waring problem for Lie groups and Chevalley groups”, Israel J. Math., 210 (2015), 81–100, arXiv: 1404.4786 | DOI | MR | Zbl

[54] Itoh M., “Invariant theory in exterior algebras and Amitsur–Levitzki type theorems”, Adv. Math., 288 (2016), 679–701, arXiv: 1404.1980 | DOI | MR | Zbl

[55] Kanel-Belov A., Grigoriev S., Elishev A., Yu J.-T., Zhang W., “Lifting of polynomial symplectomorphisms and deformation quantization”, Comm. Algebra, 46 (2018), 3926–3938, arXiv: 1707.06450 | DOI | MR | Zbl

[56] Kanel-Belov A., Karasik Y., Rowen L. H., Computational aspects of polynomial identities, v. 1, Monographs and Research Notes in Mathematics, Kemer's theorems, 2nd ed., CRC Press, Boca Raton, FL, 2016 | MR | Zbl

[57] Kanel-Belov A., Kunyavskii B., Plotkin E., “Word equations in simple groups and polynomial equations in simple algebras”, Vestnik St. Petersburg Univ. Math., 46 (2013), 3–13, arXiv: 1304.5052 | DOI | MR | Zbl

[58] Kanel-Belov A., Malev S., Rowen L., “The images of non-commutative polynomials evaluated on $2\times2$ matrices”, Proc. Amer. Math. Soc., 140 (2012), 465–478, arXiv: 1005.0191 | DOI | MR | Zbl

[59] Kanel-Belov A., Malev S., Rowen L., “The images of multilinear polynomials evaluated on $3\times 3$ matrices”, Proc. Amer. Math. Soc., 144 (2016), 7–19, arXiv: 1306.4389 | DOI | MR | Zbl

[60] Kanel-Belov A., Malev S., Rowen L., “Power-central polynomials on matrices”, J. Pure Appl. Algebra, 220 (2016), 2164–2176, arXiv: 1310.1598 | DOI | MR | Zbl

[61] Kanel-Belov A., Malev S., Rowen L., “The images of Lie polynomials evaluated on matrices”, Comm. Algebra, 45 (2017), 4801–4808, arXiv: 1506.06792 | DOI | MR | Zbl

[62] Kanel-Belov A., Rowen L. H., Computational aspects of polynomial identities, Research Notes in Mathematics, 9, A K Peters, Ltd., Wellesley, MA, 2005 | MR | Zbl

[63] Kemer A. R., “Capelli identities and nilpotence of the radical of a finitely generated PI-algebra”, Sov. Math. Dokl., 255 (1980), 750–753 | MR | Zbl

[64] Kemer A. R., “Varieties and $Z_2$-graded algebras”, Math. USSR Izv., 25 (1985), 359–374 | DOI | MR | Zbl

[65] Klimenko E., Kunyavskii B., Morita J., Plotkin E., “Word maps in Kac–Moody setting”, Toyama Math. J., 37 (2015), 25–54, arXiv: 1506.01422 | DOI | MR | Zbl

[66] Kolesnikov P. S., “The Makar-Limanov algebraically closed skew field”, Algebra Logic, 39 (2000), 378–395 | DOI | MR | Zbl

[67] Kolesnikov P. S., “On various definitions of algebraically closed skew fields”, Algebra Logic, 40 (2001), 219–230 | DOI | MR | Zbl

[68] Kulyamin V. V., “Images of graded polynomials in matrix rings over finite group algebras”, Russian Math. Surveys, 55 (2000), 345–346 | DOI | MR | Zbl

[69] Kulyamin V. V., On images of polynomials in finite matrix rings, Ph.D. Thesis, Moscow Lomonosov State University, M., 2000 | MR

[70] Larsen M., “Word maps have large image”, Israel J. Math., 139 (2004), 149–156, arXiv: math.GR/0211302 | DOI | MR | Zbl

[71] Larsen M., Shalev A., “Word maps and Waring type problems”, J. Amer. Math. Soc., 22 (2009), 437–466, arXiv: math.GR/0701334 | DOI | MR | Zbl

[72] Larsen M., Shalev A., Tiep P. H., “The Waring problem for finite simple groups”, Ann. of Math., 174 (2011), 1885–1950 | DOI | MR | Zbl

[73] Larsen M., Shalev A., Tiep P. H., “Waring problem for finite quasisimple groups”, Int. Math. Res. Not., 2013 (2013), 2323–2348, arXiv: 1107.3341 | DOI | MR | Zbl

[74] Larsen M., Tiep P. H., “A refined Waring problem for finite simple groups”, Forum Math. Sigma, 3 (2015), e6, 22 pp., arXiv: 1312.4998 | DOI | MR | Zbl

[75] Lee T.-K., Zhou Y., “Right ideals generated by an idempotent of finite rank”, Linear Algebra Appl., 431 (2009), 2118–2126 | DOI | MR | Zbl

[76] Li C., Tsui M. C., “On the images of multilinear maps of matrices over finite-dimensional division algebras”, Linear Algebra Appl., 493 (2016), 399–410 | DOI | MR | Zbl

[77] Liebeck M. W., O'Brien E. A., Shalev A., Tiep P. H., “The Ore conjecture”, J. Eur. Math. Soc. (JEMS), 12 (2010), 939–1008 | DOI | MR | Zbl

[78] Ma A., Oliva J., “On the images of Jordan polynomials evaluated over symmetric matrices”, Linear Algebra Appl., 492 (2016), 13–25 | DOI | MR | Zbl

[79] Makar-Limanov L., “Algebraically closed skew fields”, J. Algebra, 93 (1985), 117–135 | DOI | MR | Zbl

[80] Makar-Limanov L., “An example of a skew field without a trace”, Comm. Algebra, 17 (1989), 2303–2307 | DOI | MR | Zbl

[81] Malev S., “The images of non-commutative polynomials evaluated on $2\times 2$ matrices over an arbitrary field”, J. Algebra Appl., 13 (2014), 1450004, 12 pp., arXiv: 1310.8563 | DOI | MR | Zbl

[82] Malev S., “The images of noncommutative polynomials evaluated on the quaternion algebra”, J. Algebra Appl. (to appear) , arXiv: 1906.04973 | DOI | MR

[83] Malev S., Pines C., The images of multilinear non-associative polynomials evaluated on a rock-paper-scissors algebra with unit over an arbitrary field, arXiv: 2006.04517

[84] Malle G., “The proof of Ore's conjecture (after Ellers–Gordeev and Liebeck–O'Brien–Shalev–Tiep)”, Astérisque, 361, no. 1069, 2014, 325–348 | MR | Zbl

[85] Mesyan Z., “Polynomials of small degree evaluated on matrices”, Linear Multilinear Algebra, 61 (2013), 1487–1495, arXiv: 1212.1925 | DOI | MR | Zbl

[86] Procesi C., “The invariant theory of $n\times n$ matrices”, Adv. Math., 19 (1976), 306–381 | DOI | MR | Zbl

[87] Procesi C., “On the theorem of Amitsur–Levitzki”, Israel J. Math., 207 (2015), 151–154 | DOI | MR | Zbl

[88] Razmyslov Yu.P., “Finite basing of the identities of a matrix algebra of second order over a field of characteristic zero”, Algebra Logic, 12 (1973), 47–63 | DOI | MR

[89] Razmyslov Yu.P., “On a problem of Kaplansky”, Math USSR. Izv., 7 (1973), 479–496 | DOI | MR

[90] Razmyslov Yu.P., “Trace identities of full matrix algebras over a field of characteristic zero”, Math USSR. Izv., 38 (1974), 727–760 | DOI | MR

[91] Razmyslov Yu.P., Identities of algebras and their representations, Translations of Mathematical Monographs, 138, Amer. Math. Soc., Providence, RI, 1994 | DOI | MR | Zbl

[92] Rowen L. H., Polynomial identities in ring theory, Pure and Applied Mathematics, 84, Academic Press, Inc., New York–London, 1980 | MR | Zbl

[93] Rowen L. H., Graduate algebra: commutative view, Graduate Studies in Mathematics, 73, Amer. Math. Soc., Providence, RI, 2006 | DOI | MR | Zbl

[94] Rowen L. H., Graduate algebra: noncommutative view, Graduate Studies in Mathematics, 91, Amer. Math. Soc., Providence, RI, 2008 | DOI | MR | Zbl

[95] Saltman D. J., “On $p$-power central polynomials”, Proc. Amer. Math. Soc., 78 (1980), 11–13 | DOI | MR | Zbl

[96] Santulo E. A., Yasurmura F. Y., On the image of polynomials evaluated on incidence algebras: a counter-example and a solution, arXiv: 1902.08116

[97] Shalev A., “Commutators, words, conjugacy classes and character methods”, Turkish J. Math., 31, suppl. (2007), 131–148 | MR | Zbl

[98] Shalev A., “Word maps, conjugacy classes, and a noncommutative Waring-type theorem”, Ann. of Math., 170 (2009), 1383–1416 | DOI | MR | Zbl

[99] Shalev A., “Some results and problems in the theory of word maps”, Erdös Centennial, Bolyai Soc. Math. Stud., 25, János Bolyai Math. Soc., Budapest, 2013, 611–649 | DOI | MR

[100] Shirshov A. I., “Some questions in the theory of rings close to associative”, Russian Math. Surveys, 13:6 (1958), 3–20 | MR | Zbl

[101] Thom A., “Convergent sequences in discrete groups”, Canad. Math. Bull., 56 (2013), 424–433, arXiv: 1003.4093 | DOI | MR | Zbl

[102] Vitas D., Multilinear polynomials are surjective on algebras with surjective inner derivations, Preprint, 2020 | MR | Zbl

[103] Špenko Š., “On the image of a noncommutative polynomial”, J. Algebra, 377 (2013), 298–311, arXiv: 1212.4600 | DOI | MR

[104] Wang Y., “The images of multilinear polynomials on $2\times 2$ upper triangular matrix algebras”, Linear Multilinear Algebra, 67 (2019), 2366–2372 | DOI | MR | Zbl

[105] Wang Y., The image of arbitrary polynomials on $2\times 2$ upper triangular matrix algebras, Preprint, 2019 | MR

[106] Wang Y., Liu P. P., Bai J., “Correction: "The images of multilinear polynomials on $2\times 2$ upper triangular matrix algebras"”, Linear Multilinear Algebra, 67:11 (2019), i–vi | DOI | MR | Zbl

[107] Watkins W., “Linear maps that preserve commuting pairs of matrices”, Linear Algebra Appl., 14 (1976), 29–35 | DOI | MR | Zbl

[108] Zelmanov E., “Infinite algebras and pro-$p$ groups”, Infinite groups: geometric, combinatorial and dynamical aspects, Progr. Math., 248, Birkhäuser, Basel, 2005, 403–413 | DOI | MR | Zbl

[109] Zhevlakov K. A., Slinko A. M., Shestakov I. P., Shirshov A. I., Rings close to associative, Nauka, M., 1978 | MR

[110] Zubkov A. N., “Non-abelian free pro-$p$-groups cannot be represented by 2-by-2 matrices”, Sib. Math. J., 28 (1987), 742–747 | DOI | MR | Zbl