论文标题
积分量子簇结构
Integral quantum cluster structures
论文作者
论文摘要
我们证明了一个通用定理，用于构建$ {\ Mathbb {z}}} [q^{\ pm 1/2}] $上的整体量子群集代数，而不是在温和条件下，量子nilpotent代数的积分形式始终具有整体量子群集量群集Algebra结构。然后，这些代数被证明是对应的上量子群集代数的同构，在$ {\ mathbb {z}}} [q^{\ pm 1/2}] $上再次定义。以前，这仅以无环量子簇代数而闻名。该定理用于证明，对于每个对称的kac-moody代数$ {\ mathfrak {g}} $和weyl group元素$ w $，双义式形式$ a_q（{\ mathfrak {n}}}}} _+（w）_ {量子单位单元格具有$ a_q（{\ mathfrak {n}} _+（w））_ {\ Mathbb {z} [q^{q^{\ pm 1}} \ otimes _ { Q^{\ pm 1/2}] $是$ {\ Mathbb {z}}}量子代数的同构，[q^{q^{\ pm 1/2}] $，对相应的上量子群集代数超过$ {\ mathbb {\ mathbb {z}} [q^q^^= {
We prove a general theorem for constructing integral quantum cluster algebras over ${\mathbb{Z}}[q^{\pm 1/2}]$, namely that under mild conditions the integral forms of quantum nilpotent algebras always possess integral quantum cluster algebra structures. These algebras are then shown to be isomorphic to the corresponding upper quantum cluster algebras, again defined over ${\mathbb{Z}}[q^{\pm 1/2}]$. Previously, this was only known for acyclic quantum cluster algebras. The theorem is applied to prove that for every symmetrizable Kac-Moody algebra ${\mathfrak{g}}$ and Weyl group element $w$, the dual canonical form $A_q({\mathfrak{n}}_+(w))_{\mathbb{Z}[q^{\pm 1}]}$ of the corresponding quantum unipotent cell has the property that $A_q( {\mathfrak{n}}_+(w))_{\mathbb{Z}[q^{\pm 1}]} \otimes_{\mathbb{Z}[q^{ \pm 1}]} {\mathbb{Z}}[ q^{\pm 1/2}]$ is isomorphic to a quantum cluster algebra over ${\mathbb{Z}}[q^{\pm 1/2}]$ and to the corresponding upper quantum cluster algebra over ${\mathbb{Z}}[q^{\pm 1/2}]$.
