The proof of Kontsevich's periodicity conjecture on noncommutative birational transformations

<p style="margin: 0px 0px 1em; padding: 0px; border: 0px; font-variant-numeric: inherit; font-stretch: inherit; font-size: 0.875em; line-height: 1.5em; font-family: "Helvetica Neue", Helvetica, Arial, sans-serif; vertical-align: baseline; color: rgb(34, 34, 34); background-color: rgb(255, 255, 255);">For an arbitrary associative unital ring RR, let J1J1 and J2J2 be the following noncommutative, birational, partly defined involutions on the set M3(R)M3(R) of 3×33×3 matrices over RR: J1(M)=M−1J1(M)=M−1 (the usual matrix inverse) and J2(M)jk=(Mkj)−1J2(M)jk=(Mkj)−1 (the transpose of the Hadamard inverse).</p><p style="margin: 0px 0px 1em; padding: 0px; border: 0px; font-variant-numeric: inherit; font-stretch: inherit; font-size: 0.875em; line-height: 1.5em; font-family: "Helvetica Neue", Helvetica, Arial, sans-serif; vertical-align: baseline; color: rgb(34, 34, 34); background-color: rgb(255, 255, 255);">We prove the surprising conjecture by Kontsevich that (J2∘J1)3(J2∘J1)3 is the identity map modulo the DiagL×DiagRDiagL×DiagR action (D1,D2)(M)=D−11MD2(D1,D2)(M)=D1−1MD2 of pairs of invertible diagonal matrices. That is, we show that, for each MM in the domain where (J2∘J1)3(J2∘J1)3 is defined, there are invertible diagonal 3×33×3 matrices D1=D1(M)D1=D1(M) and D2=D2(M)D2=D2(M) such that (J2∘J1)3(M)=D−11MD2(J2∘J1)3(M)=D1−1MD2.</p>