The proof of Kontsevich's periodicity conjecture on noncommutative birational transformations

For an arbitrary associative unital ring RR, let J1J1 and J2J2 be the following noncommutative, birational, partly defined involutions on the set M3(R)M3(R) of 3×33×3 matrices over RR: J1(M)=M−1J1(M)=M−1 (the usual matrix inverse) and J2(M)jk=(Mkj)−1J2(M)jk=(Mkj)−1 (the transpose of the Hadamard inverse).

We prove the surprising conjecture by Kontsevich that (J2∘J1)3(J2∘J1)3 is the identity map modulo the DiagL×DiagRDiagL×DiagR action (D1,D2)(M)=D−11MD2(D1,D2)(M)=D1−1MD2 of pairs of invertible diagonal matrices. That is, we show that, for each MM in the domain where (J2∘J1)3(J2∘J1)3 is defined, there are invertible diagonal 3×33×3 matrices D1=D1(M)D1=D1(M) and D2=D2(M)D2=D2(M) such that (J2∘J1)3(M)=D−11MD2(J2∘J1)3(M)=D1−1MD2.

Non-trivial stably free modules over crossed products

We consider the class of crossed products of noetherian domains with universal enveloping algebras of Lie algebras. For algebras from this class we give a sufficient condition for the existence of projective non-free modules. This class includes Weyl algebras and universal envelopings of Lie algebras, for which this question, known as noncommutative Serre's problem, was extensively studied before. It turns out that the method of lifting of non-trivial stably free modules from simple Ore extensions can be applied to crossed products after an appropriate choice of filtration. The motivating examples of crossed products are provided by the class of RIT algebras, originating in non-equilibrium physics.

Iterative Multiuser Detection and Decoding for DS-CDMA System With Space-Time Linear Dispersion

This paper considers a Q-ary orthogonal direct-sequence code-division multiple-access (DS-CDMA) system with high-rate space-time linear dispersion codes (LDCs) in time-varying Rayleigh fading multiple-input-multiple-output (MIMO) channels. We propose a joint multiuser detection, LDC decoding, Q-ary demodulation, and channel-decoding algorithm and apply the turbo processing principle to improve system performance in an iterative fashion. The proposed iterative scheme demonstrates faster convergence and superior performance compared with the V-BLAST-based DS-CDMA system and is shown to approach the single-user performance bound. We also show that the CDMA system is able to exploit the time diversity offered by the LDCS in rapid-fading channels.