## The proof of the Kontsevich periodicity conjecture on noncommutative birational transformations.(English)Zbl 1334.16028

Summary: 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 surprising conjecture by Kontsevich that $$(J_2\circ J_1)^3$$ is the identity map modulo the $$\mathrm{Diag}_L\times\mathrm{Diag}_R$$ 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 $$(J_2\circ J_1)^3(M)=D_1^{-1}MD_2$$.

### MSC:

 16S38 Rings arising from noncommutative algebraic geometry 16S50 Endomorphism rings; matrix rings 15A15 Determinants, permanents, traces, other special matrix functions 15A09 Theory of matrix inversion and generalized inverses 37F10 Dynamics of complex polynomials, rational maps, entire and meromorphic functions; Fatou and Julia sets 13F60 Cluster algebras 16S85 Associative rings of fractions and localizations
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