Quantization of deformed cluster Poisson varieties

Abstract

Fock and Goncharov described a quantization of cluster X -varieties (also known as cluster Poisson varieties) in [FG09]. Meanwhile, families of deformations of cluster X -varieties were introduced in [BFMN18]. In this paper we show that the two constructions are compatible-we extend the Fock-Goncharov quantization of X -varieties to the families of [BFMN18]. As a corollary, we obtain that these families and each of their fibers have Poisson structures. We relate this construction to the Berenstein-Zelevinsky quantization of A-varieties ([BZ05]). Finally, inspired by the counter-example to quantum positivity of the quantum greedy basis in [LLRZ14], we compute a counter-example to quantum positivity of the quantum theta basis.

Pentagon relation in quantum cluster scattering diagrams

“…Having this development in mind, in this paper we formulate the pentagon identity among the quantum dilogarithm elements, which is the counterpart of the one for the dilogarithm elements for a CSD in [Nak21a]. As an application, we show the nonpositivity of a certain class of nonskew-symmetric QCSDs (Theorem 5.6 and Corollary 5.7), which is related to the results of [LLRZ14,DM21,CFMM20]. Also, we explicitly present various consistency relations for QCSDs of rank 2 completely or up to some degree, many of which are new in the literature.…”

“…Observe that this is indeed the automorphism part of the Fock-Goncharov decomposition of mutations of the quantum x-variables with principal coefficients in [BZ05]. See also [Man15,DM21,CFMM20] for an alternative approach, where the same representation is described through the adjoint action of the quantum dilogarithm.…”

“…It is known that in the classical case the greedy basis and the theta basis coincide [CGM + 17]. Also, the above nonpositivity of the quantum greedy elements was explained in view of the nonpositivity of the wall functions in the corresponding QCSD [CFMM20] for the case (d 1 , d 2 ) = (2, 3). We see that there is an intriguing discrepancy of the positivity property between CSDs and QCSDs especially in the nonskew-symmetric case.…”

“…(d 1 , d 2 ) = (2, 3). This is the case where the nonpositivity of a quantum greedy element is presented in [LLRZ14], and further examined in view of a scattering diagram in [CFMM20]. We have…”