The proof of the Kontsevich periodicity conjecture on noncommutative birational transformations. (English) Zbl 1334.16028

Summary: For an arbitrary associative unital ring \(R\), let \(J_1\) and \(J_2\) be the following noncommutative, birational, partly defined involutions on the set \(M_3(R)\) of \(3\times 3\) matrices over \(R\): \(J_1(M)=M^{-1}\) (the usual matrix inverse) and \(J_2(M)_{jk}=(M_{kj})^{-1}\) (the transpose of the Hadamard inverse).
We prove the surprising conjecture by Kontsevich that \((J_2\circ J_1)^3\) is the identity map modulo the \(\mathrm{Diag}_L\times\mathrm{Diag}_R\) action \((D_1,D_2)(M)=D_1^{-1}MD_2\) of pairs of invertible diagonal matrices. That is, we show that, for each \(M\) in the domain where \((J_2\circ J_1)^3\) is defined, there are invertible diagonal \(3\times 3\) matrices \(D_1=D_1(M)\) and \(D_2=D_2(M)\) such that \((J_2\circ J_1)^3(M)=D_1^{-1}MD_2\).


16S38 Rings arising from noncommutative algebraic geometry
16S50 Endomorphism rings; matrix rings
15A15 Determinants, permanents, traces, other special matrix functions
15A09 Theory of matrix inversion and generalized inverses
37F10 Dynamics of complex polynomials, rational maps, entire and meromorphic functions; Fatou and Julia sets
13F60 Cluster algebras
16S85 Associative rings of fractions and localizations
Full Text: DOI arXiv Euclid Link


[1] V. I. Arnold, “Dynamics of the complexity of intersections” in Analysis, et cetera , Academic Press, New York, 1990. · Zbl 0782.54020 · doi:10.1007/BF01236277
[2] A. Berenstein and V. Retakh, A short proof of Kontsevich’s cluster conjecture , C. R. Math. Acad. Sci. Paris 349 (2011), 119-122. · Zbl 1266.16026 · doi:10.1016/j.crma.2011.01.004
[3] P. M. Cohn and C. Reutenauer, On the construction of the free field , Internat. J. Algebra Comput. 9 (1999), 307-323. · Zbl 1040.16015 · doi:10.1142/S0218196799000205
[4] I. Dolgachev and D. Ortland, Point Sets in Projective Spaces and Theta Functions , Astérisque 165 , Soc. Math. France, Paris, 1988. · Zbl 0685.14029
[5] I. Gelfand, S. Gelfand, V. Retakh, and R. L. Wilson, Quasideterminants , Adv. Math. 193 (2005), 56-141. · Zbl 1079.15007 · doi:10.1016/j.aim.2004.03.018
[6] C. Kassel and C. Reutenauer, Algebraicity of the zeta function associated to a matrix over a free group algebra , Algebra Number Theory 8 (2014), 497-511. · Zbl 1302.14006 · doi:10.2140/ant.2014.8.497
[7] M. Kontsevich, Noncommutative identities , preprint, [math.RA]. arXiv:1109.2469v1
[8] A. P. Veselov, Growth and integrability in the dynamics of mappings , Commun. Math. Phys. 145 (1992), 181-193. · Zbl 0751.58034 · doi:10.1007/BF02099285
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.