Quantum generalized cluster algebras and quantum dilogarithms of higher degrees

Abstract

Abstract. We extend the notion of the quantization of the coefficients of the ordinary cluster algebras to the generalized cluster algebras by Chekhov and Shapiro. In parallel to the ordinary case, it is tightly integrated with certain generalizations of the ordinary quantum dilogarithm, which we call the quantum dilogarithms of higher degrees. As an application, we derive the identities of these generalized quantum dilogarithms associated with any period of quantum Y -seeds. IntroductionThe generalized cluster algebras were introduced by Chekhov and Shapiro [CS14]. In this note we demonstrate that the notion of quantum cluster algebras is also extended to the generalized ones. To be more precise, there are two kinds of formulations of quantum cluster algebras, the one quantizing the cluster variables by [BZ05] and the one quantizing the coefficients by [FG09a,FG09b], and it is known that they are closely related to each other. Here, we concentrate on the latter one. As shown by [FG09a,FG09b], in the ordinary case, the quantization of the coefficients is tightly integrated with the quantum dilogarithm [FV93,FK94]. Similarly, in the generalized case, it is tightly integrated with certain generalizations of the quantum dilogarithm, which we call the quantum dilogarithms of higher degrees. As an application, we derive the identities of these generalized quantum dilogarithms associated with any period of quantum Y -seeds, which are also parallel to the ones in the ordinary case.The main message of this note is that the fundamental (and perhaps all) features of the quantum cluster algebras are also extended to the generalized ones.2010 Mathematics Subject Classification. 13F60.

Rogers dilogarithms of higher degree and generalized cluster algebras

“…Here we complete the picture by showing how the classical dilogarithm identity of higher degree in Theorem 4.8 is obtained from its quantum counterpart in [Nak15a,Theorem 4.1]. This is a generalization of the argument in [KN11] for the ordinary cluster algebras with skew-symmetric exchange matrices.…”

“…Here we follow and generalize the calculations especially in Section 4 and Appendix A of [KN11]. Since this is a rather complicated subject, we try to write it in a selfcontained way at a reasonable level, but not completely, and we ask the readers to refer to [KN11] (and also [Nak15a]) for further details.…”

“…In fact, it is a very natural generalization so that all nice properties of cluster algebras are shown or expected to hold [Rup13], [Nak15b]. Naturally one can further define the quantum generalized cluster algebras as well [Nak15a]. For ordinary quantum cluster algebras, the quantum dilogarithm (1.2) plays a key role to control their mutations [FG09a], [FG09b].…”

“…(II'). There is a quantum dilogarithm identity of higher degree associated with any period of seeds of a quantum generalized cluster algebra [Nak15a].…”

“…In fact, a classical counterpart of the function (1.4) was already introduced in [Nak15a], and it looks as follows: (This notation is used only here. )…”