量子α-行列式与q-变形超几何多项式
Quantum alpha-determinants and q-deformed hypergeometric polynomials
摘要：
量子$α$ -行列式被定义为量子行列式的参数变形。研究了由量子$α$ -行列式的幂生成的量子矩阵代数$\mathcal{A}_q(\mathrm{Mat}_2)$的循环$\mathcal{U}_q(\mathfrak{sl}_2)$ -子模.对于这样一个循环模，存在一组多项式，它们以下列方式描述它的不可约分解：（i）每个多项式对应于某个不可约$\mathcal{U}_q(\mathfrak{sl}_2)$ -模，（ii）如果参数是相应多项式的根，则循环模包含一个不可约子模。这些多项式作为超几何多项式的$q$ -变形给出。这是我们以前工作[K. Kimoto，S. Matsumoto和M. Wakayama，α-行列式循环模和雅可比多项式，出现在美国数学学会。
英文摘要：
The quantum $α$-determinant is defined as a parametric deformation of the quantum determinant. We investigate the cyclic $\mathcal{U}_q(\mathfrak{sl}_2)$-submodules of the quantum matrix algebra $\mathcal{A}_q(\mathrm{Mat}_2)$ generated by the powers of the quantum $α$-determinant. For such a cyclic module, there exists a collection of polynomials which describe the irreducible decomposition of it in the following manner: (i) each polynomial corresponds to a certain irreducible $\mathcal{U}_q(\mathfrak{sl}_2)$-module, (ii) the cyclic module contains an irreducible submodule if the parameter is a root of the corresponding polynomial. These polynomials are given as a $q$-deformation of the hypergeometric polynomials. This is a quantum analogue of the result obtained in our previous work [K. Kimoto, S. Matsumoto and M. Wakayama, Alpha-determinant cyclic modules and Jacobi polynomials, to appear in Trans. Amer. Math. Soc.].
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
