The proof of the Kontsevich periodicity conjecture on noncommutative birational transformations

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 $M_3\left(R\right)$ of $3\times 3$ matrices over $R$: $J_1\left(M\right)=M^{-1}$ (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 ${Diag}_L\times {Diag}_R$ action $(D_1,D_2)\left(M\right)=D_1^{-1}M{D}_2$ 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 $D_1=D_1\left(M\right)$ and $D_2=D_2\left(M\right)$ such that $(J_2\circ J_1{)}^3(M)=D_1^{-1}M{D}_2$.