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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 \((J_2\circ J_1)^3(M)=D_1^{-1}MD_2\).

MSC:

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
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