The proof of the Kontsevich periodicity conjecture on noncommutative birational transformations

TL;DR

This paper proves Kontsevich's conjecture that a specific composition of noncommutative birational involutions on 3x3 matrices over any ring has order three up to diagonal conjugation, revealing a deep symmetry in noncommutative algebra.

Contribution

We establish that the third power of the composition of two particular noncommutative involutions equals the identity modulo diagonal conjugation, confirming a conjecture by Kontsevich.

Findings

The composition (J2∘J1)^3 acts as the identity modulo diagonal matrices.

The result holds for matrices over any associative unital ring.

The proof confirms a key symmetry in 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 $M_3(R)$ 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 following surprising conjecture by Kontsevich saying that $(J_2\circ J_1)^3$ is the identity map modulo the ${\rm Diag}_{L} \times \rm{Diag}_R$ 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 $(J_2\circ J_1)^3(M)=D_1^{-1}MD_2.$