We construct “quantum theta bases,” extending the set of quantum cluster monomials, for various versions of skew-symmetric quantum cluster algebras. These bases consist precisely of the indecomposable universally positive elements of the algebras they generate, and the structure constants for their multiplication are Laurent polynomials in the quantum parameter with non-negative integer coefficients, proving the quantum strong cluster positivity conjecture for these algebras. The classical limits recover the theta bases considered by Gross–Hacking–Keel–Kontsevich (J Am Math Soc 31(2):497–608, 2018). Our approach combines the scattering diagram techniques used in loc. cit. with the Donaldson–Thomas theory of quivers.

我们构建了“量子 theta 基”，扩展了量子簇单项式的集合，用于各种版本的斜对称量子簇代数。这些基正是由它们生成的代数的不可分解的普遍正元素组成，它们相乘的结构常数是量子参数中具有非负整数系数的洛朗多项式，证明了这些代数的量子强群正性猜想。经典极限恢复了 Gross-Hacking-Keel-Kontsevich (J Am Math Soc 31(2):497-608, 2018) 考虑的 theta 基数。我们的方法结合了 loc 中使用的散射图技术。引用。与唐纳森-托马斯的箭袋理论。