AI 中文总结
本文为拟三角剖分带孔曲面的约化Lê--Roger--Yang skein代数建立广义量子簇代数结构，定义并应用Frobenius映射，确定相关代数的中心并计算其作为中心模的秩。
AI 中文摘要
我们建立了由Bai--Chen--Ding--Xu定义的广义量子簇代数结构，该结构适用于每个拟三角剖分带孔曲面的约化Lê--Roger--Yang skein代数。我们在单位根处定义了广义量子簇代数的Frobenius映射，并利用这些映射连同簇代数结构，为所有带孔曲面的（约化）Lê--Roger--Yang skein代数构造了Frobenius映射。作为Frobenius映射的应用，我们确定了单位根处Muller--Roger--Yang skein代数和约化Lê--Roger--Yang skein代数的中心，并计算了它们作为其中心上的模的秩。
英文摘要
We establish a generalized quantum cluster algebra structure, defined by Bai--Chen--Ding--Xu, on the reduced Lê--Roger--Yang skein algebra of every quasi-triangulable punctured surface. We define Frobenius maps for generalized quantum cluster algebras at roots of unity and use these maps, together with the cluster algebra structure, to construct Frobenius maps for the (reduced) Lê--Roger--Yang skein algebras of all punctured surfaces. As applications of the Frobenius map, we determine the centers of both the Muller--Roger--Yang skein algebras and the reduced Lê--Roger--Yang skein algebras at roots of unity, and compute their ranks as modules over their centers.
Comments57 pages
