The images of non-commutative polynomials evaluated on 2×2 matrices

Kanel-Belov, Alexey; Malev, Sergey; Rowen, Louis

doi:10.1090/S0002-9939-2011-10963-8

The images of non-commutative polynomials evaluated on $2\times 2$ matrices
HTML articles powered by AMS MathViewer

by Alexey Kanel-Belov, Sergey Malev and Louis Rowen PDF

Proc. Amer. Math. Soc. 140 (2012), 465-478 Request permission

Abstract:

Let $p$ be a multilinear polynomial in several non-commuting variables with coefficients in a quadratically closed field $K$ of any characteristic. It has been conjectured that for any $n$, 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 $sl_n(K)$ of matrices of trace 0, or all of $M_n(K)$. We prove the conjecture for $n=2$, and show that although the analogous assertion fails for completely homogeneous polynomials, one can salvage the conjecture in this case by including the set of all non-nilpotent matrices of trace zero and also permitting dense subsets of $M_n(K)$.

References

A. A. Albert and Benjamin Muckenhoupt, On matrices of trace zeros, Michigan Math. J. 4 (1957), 1–3. MR 83961, DOI 10.1307/mmj/1028990168
A. S. Amitsur and J. Levitzki, Minimal identities for algebras, Proc. Amer. Math. Soc. 1 (1950), 449–463. MR 36751, DOI 10.1090/S0002-9939-1950-0036751-9
A. Ya. Belov, Counterexamples to the Specht problem, Mat. Sb. 191 (2000), no. 3, 13–24 (Russian, with Russian summary); English transl., Sb. Math. 191 (2000), no. 3-4, 329–340. MR 1773251, DOI 10.1070/SM2000v191n03ABEH000460
A. Ya. Belov, V. V. Borisenko, and V. N. Latyshev, Monomial algebras, J. Math. Sci. (New York) 87 (1997), no. 3, 3463–3575. Algebra, 4. MR 1604202, DOI 10.1007/BF02355446
Matej Brešar and Igor Klep, Values of noncommutative polynomials, Lie skew-ideals and tracial Nullstellensätze, Math. Res. Lett. 16 (2009), no. 4, 605–626. MR 2525028, DOI 10.4310/MRL.2009.v16.n4.a5
A. Borel, On free subgroups of semisimple groups, Enseign. Math. (2) 29 (1983), no. 1-2, 151–164. MR 702738
Alexei Kanel-Belov and Louis Halle Rowen, Computational aspects of polynomial identities, Research Notes in Mathematics, vol. 9, A K Peters, Ltd., Wellesley, MA, 2005. MR 2124127, DOI 10.1201/9781439863725
Chen-Lian Chuang, On ranges of polynomials in finite matrix rings, Proc. Amer. Math. Soc. 110 (1990), no. 2, 293–302. MR 1027090, DOI 10.1090/S0002-9939-1990-1027090-3
Stephen Donkin, Invariants of several matrices, Invent. Math. 110 (1992), no. 2, 389–401. MR 1185589, DOI 10.1007/BF01231338
Vesselin Drensky and Edward Formanek, Polynomial identity rings, Advanced Courses in Mathematics. CRM Barcelona, Birkhäuser Verlag, Basel, 2004. MR 2064082, DOI 10.1007/978-3-0348-7934-7
Dnestrovskaya tetrad′, 4th ed., Rossiĭskaya Akademiya Nauk Sibirskoe Otdelenie, Institut Matematiki im. S. L. Soboleva, Novosibirsk, 1993 (Russian). Nereshennye problemy teorii kolets i moduleĭ. [Unsolved problems in the theory of rings and modules]; Compiled by V. T. Filippov, V. K. Kharchenko and I. P. Shestakov. MR 1310114
Vesselin Drensky and Giulia Maria Piacentini Cattaneo, A central polynomial of low degree for $4\times 4$ matrices, J. Algebra 168 (1994), no. 2, 469–478. MR 1292776, DOI 10.1006/jabr.1994.1240
Edward Formanek, Central polynomials for matrix rings, J. Algebra 23 (1972), 129–132. MR 302689, DOI 10.1016/0021-8693(72)90050-6
Edward Formanek, The polynomial identities and invariants of $n\times n$ matrices, CBMS Regional Conference Series in Mathematics, vol. 78, Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 1991. MR 1088481, DOI 10.1090/cbms/078
A. È. Guterman and B. Kuz′ma, Mappings that preserve the zeros of matrix polynomials, Dokl. Akad. Nauk 427 (2009), no. 3, 300–302 (Russian); English transl., Dokl. Math. 80 (2009), no. 1, 508–510. MR 2573059, DOI 10.1134/S1064562409040152
Patrick Halpin, Central and weak identities for matrices, Comm. Algebra 11 (1983), no. 19, 2237–2248. MR 714201, DOI 10.1080/00927878308822961
H. Helling, Eine Kennzeichnung von Charakteren auf Gruppen und assoziativen Algebren, Comm. Algebra 1 (1974), 491–501 (German). MR 347880, DOI 10.1080/00927877408548718
I. N. Herstein, On the Lie structure of an associative ring, J. Algebra 14 (1970), 561–571. MR 255610, DOI 10.1016/0021-8693(70)90103-1
A. R. Kemer, Capelli identities and nilpotency of the radical of finitely generated PI-algebra, Dokl. Akad. Nauk SSSR 255 (1980), no. 4, 793–797 (Russian). MR 600746
A. R. Kemer, Varieties and $Z_2$-graded algebras, Izv. Akad. Nauk SSSR Ser. Mat. 48 (1984), no. 5, 1042–1059 (Russian). MR 764308
V. V. Kulyamin, Images of graded polynomials in matrix rings over finite group algebras, Uspekhi Mat. Nauk 55 (2000), no. 2(332), 141–142 (Russian); English transl., Russian Math. Surveys 55 (2000), no. 2, 345–346. MR 1781072, DOI 10.1070/rm2000v055n02ABEH000278
Kulyamin, V.V. On images of polynomials in finite matrix rings, Thes. Cand. Phys.-Math. Sci., Moscow Lomonosov State University, Moscow (2000).
Michael Larsen, Word maps have large image, Israel J. Math. 139 (2004), 149–156. MR 2041227, DOI 10.1007/BF02787545
Michael Larsen and Aner Shalev, Word maps and Waring type problems, J. Amer. Math. Soc. 22 (2009), no. 2, 437–466. MR 2476780, DOI 10.1090/S0894-0347-08-00615-2
Tsiu-Kwen Lee and Yiqiang Zhou, Right ideals generated by an idempotent of finite rank, Linear Algebra Appl. 431 (2009), no. 11, 2118–2126. MR 2567818, DOI 10.1016/j.laa.2009.07.005
C. Procesi, The invariant theory of $n\times n$ matrices, Advances in Math. 19 (1976), no. 3, 306–381. MR 419491, DOI 10.1016/0001-8708(76)90027-X
Ju. P. Razmyslov, A certain problem of Kaplansky, Izv. Akad. Nauk SSSR Ser. Mat. 37 (1973), 483–501 (Russian). MR 0338063
Ju. P. Razmyslov, The existence of a finite basis for the identities of the matrix algebra of order two over a field of characteristic zero, Algebra i Logika 12 (1973), 83–113, 121 (Russian). MR 0340348
Ju. P. Razmyslov, Identities with trace in full matrix algebras over a field of characteristic zero, Izv. Akad. Nauk SSSR Ser. Mat. 38 (1974), 723–756 (Russian). MR 0506414
Louis Halle Rowen, Polynomial identities in ring theory, Pure and Applied Mathematics, vol. 84, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1980. MR 576061
Aner Shalev, Word maps, conjugacy classes, and a noncommutative Waring-type theorem, Ann. of Math. (2) 170 (2009), no. 3, 1383–1416. MR 2600876, DOI 10.4007/annals.2009.170.1383
William Watkins, Linear maps that preserve commuting pairs of matrices, Linear Algebra Appl. 14 (1976), no. 1, 29–35. MR 480574, DOI 10.1016/0024-3795(76)90060-4
Efim Zelmanov, Infinite algebras and pro-$p$ groups, Infinite groups: geometric, combinatorial and dynamical aspects, Progr. Math., vol. 248, Birkhäuser, Basel, 2005, pp. 403–413. MR 2195460, DOI 10.1007/3-7643-7447-0_{1}1
A. N. Zubkov, On a generalization of the Razmyslov-Procesi theorem, Algebra i Logika 35 (1996), no. 4, 433–457, 498 (Russian, with Russian summary); English transl., Algebra and Logic 35 (1996), no. 4, 241–254. MR 1444429, DOI 10.1007/BF02367026