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1 October 2015 The proof of the Kontsevich periodicity conjecture on noncommutative birational transformations
Natalia Iyudu, Stanislav Shkarin
Duke Math. J. 164(13): 2539-2575 (1 October 2015). DOI: 10.1215/00127094-3146603

Abstract

For an arbitrary associative unital ring R, let J1 and J2 be the following noncommutative, birational, partly defined involutions on the set M3(R) of 3×3 matrices over R: J1(M)=M1 (the usual matrix inverse) and J2(M)jk=(Mkj)1 (the transpose of the Hadamard inverse).

We prove the surprising conjecture by Kontsevich that (J2J1)3 is the identity map modulo the DiagL×DiagR action (D1,D2)(M)=D11MD2 of pairs of invertible diagonal matrices. That is, we show that, for each M in the domain where (J2J1)3 is defined, there are invertible diagonal 3×3 matrices D1=D1(M) and D2=D2(M) such that (J2J1)3(M)=D11MD2.

Citation

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Natalia Iyudu. Stanislav Shkarin. "The proof of the Kontsevich periodicity conjecture on noncommutative birational transformations." Duke Math. J. 164 (13) 2539 - 2575, 1 October 2015. https://doi.org/10.1215/00127094-3146603

Information

Received: 7 May 2014; Revised: 11 November 2014; Published: 1 October 2015
First available in Project Euclid: 5 October 2015

zbMATH: 1334.16028
MathSciNet: MR3405593
Digital Object Identifier: 10.1215/00127094-3146603

Subjects:
Primary: 16S38
Secondary: 16S50, 16S85

Rights: Copyright © 2015 Duke University Press

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Vol.164 • No. 13 • 1 October 2015
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