The proof of the Kontsevich periodicity conjecture on noncommutative birational transformations

Natalia Iyudu; Stanislav Shkarin

doi:10.1215/00127094-3146603

paper

The proof of the Kontsevich periodicity conjecture on noncommutative birational transformations

arXiv:1305.1965 · doi:10.1215/00127094-3146603

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 $J_{2} (M)_{j k} = (M_{k j})^{- 1}$ (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 $Diag_{L} \times Diag_{R}$ action $(D_{1}, D_{2}) (M) = D_{1}^{- 1} M D_{2}$ 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 $D_{1} = D_{1} (M)$ and $D_{2} = D_{2} (M)$ such that $(J_{2} \circ J_{1})^{3} (M) = D_{1}^{- 1} M D_{2} .$

28 pages

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