The proof of the Kontsevich periodicity conjecture on noncommutative birational transformations

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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)=M⁻¹ (the usual matrix inverse) and J₂(M)jk=(Mkj)⁻¹ (the transpose of the Hadamard inverse). We prove the following surprising conjecture by Kontsevich saying that (J₂∘ J₁)³ is the identity map modulo the Diagₗ × Diagᵣ action (D₁,D₂)(M)=D₁⁻¹MD₂ of pairs of invertible diagonal matrices. That is, we show that for each M in the domain where (J₂∘ J₁)³ is defined, there are invertible diagonal 3× 3 matrices D₁=D₁(M) and D₂=D₂(M) such that (J₂∘ J₁)³(M)=D₁⁻¹MD₂.