For an arbitrary associative unital ring , let and be the following noncommutative, birational, partly defined involutions on the set of matrices over : (the usual matrix inverse) and (the transpose of the Hadamard inverse).
We prove the surprising conjecture by Kontsevich that is the identity map modulo the action of pairs of invertible diagonal matrices. That is, we show that, for each in the domain where is defined, there are invertible diagonal matrices and such that .
"The proof of the Kontsevich periodicity conjecture on noncommutative birational transformations." Duke Math. J. 164 (13) 2539 - 2575, 1 October 2015. https://doi.org/10.1215/00127094-3146603