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The proof of the Kontsevich periodicity conjecture on noncommutative birational transformations

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http://projecteuclid.org/euclid.dmj/1444051068
Original languageEnglish
Pages (from-to)2539-2575
JournalDuke Mathematical Journal
Volume164
Issue number13
DOIs
Publication statusPublished - Oct 2015

Abstract

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 3×3 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 following surprising conjecture by Kontsevich saying that (J 2 ∘J 1 ) 3 is the identity map modulo the Diag L ×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 ∘J 1 ) 3 is defined, there are invertible diagonal 3×3 matrices D 1 =D 1 (M) and D 2 =D 2 (M) such that (J 2 ∘J 1 ) 3 (M)=D −1 1 MD 2 .

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