A Quantum Analog of Generalized Cluster Algebras

Abstract

We define a quantum analogue of a class of generalized cluster algebras which can be viewed as a generalization of quantum cluster algebras defined in [2]. In the case of rank two, we extend some structural results from the classical theory of generalized cluster algebras obtained in [3][12] to the quantum case.

Rank two non-commutative Laurent phenomenon and pseudo-positivity

“…We give a direct proof here, however the combinatorial construction below provides an alternate proof. See [1] for a proof of this result in the special case when P 1 = P 1 and P 2 = P 2 .…”

“…Note that these are again polynomials of the same form. For notational convenience, for k ∈ Z we define (1) P k (z) = p k,0 + p k,1 z + · · · + p k,…”

Generalized quantum cluster algebras: the Laurent phenomenon and upper bounds

“…Generalized quantum cluster algebras introduced in [1] naturally extend the definition of quantum cluster algebras and can be considered as quantum analogues of generalized cluster algebras of geometric types [7]. We proved in [1] that the Laurent phenomenon holds only in these generalized quantum cluster algebras of rank two.…”

“…Generalized quantum cluster algebras introduced in [1] naturally extend the definition of quantum cluster algebras and can be considered as quantum analogues of generalized cluster algebras of geometric types [7]. We proved in [1] that the Laurent phenomenon holds only in these generalized quantum cluster algebras of rank two. In this paper, we obtain the Laurent phenomenon in all generalized quantum cluster algebras of geometric types, i.e., all generalized cluster variables belong to an intersection of certain (infinitely many) rings of Laurent polynomials which is called generalized quantum upper cluster algebras.…”

“…, j} for i, j ∈ Z and i < j. At first, we recall the definition of the generalized quantum cluster algebras in [1]. b kj λ ki = δ ij d j , it follows that for the standard column vector e l we have…”