Letters in Mathematical Physics ( IF 1.3 ) Pub Date : 2021-06-18 , DOI: 10.1007/s11005-021-01422-0 Hau-Wen Huang
Assume that \({\mathbb {F}}\) is an algebraically closed field and let q denote a nonzero scalar in \({\mathbb {F}}\) that is not a root of unity. The universal Askey–Wilson algebra \(\triangle _q\) is a unital associative \({\mathbb {F}}\)-algebra defined by generators and relations. The generators are A, B, C and the relations state that each of
$$\begin{aligned} A+\frac{q BC-q^{-1} CB}{q^2-q^{-2}}, \qquad B+\frac{q CA-q^{-1} AC}{q^2-q^{-2}}, \qquad C+\frac{q AB-q^{-1} BA}{q^2-q^{-2}} \end{aligned}$$is central in \(\triangle _q\). The universal DAHA (double affine Hecke algebra) \({\mathfrak {H}}_q\) of type \((C_1^\vee ,C_1)\) is a unital associative \({\mathbb {F}}\)-algebra generated by \(\{t_i^{\pm 1}\}_{i=0}^3\), and the relations state that
$$\begin{aligned}&t_it_i^{-1}=t_i^{-1} t_i=1 \quad \hbox {for all } i=0,1,2,3;\\&t_i+t_i^{-1} \hbox { is central} \quad \hbox {for all } i=0,1,2,3; \\&\qquad \qquad \qquad t_0t_1t_2t_3=q^{-1}. \end{aligned}$$Each \({\mathfrak {H}}_q\)-module is a \(\triangle _q\)-module by pulling back via the injection \(\triangle _q\rightarrow {\mathfrak {H}}_q\) given by
$$\begin{aligned} A\mapsto & {} t_1 t_0+(t_1 t_0)^{-1}, \\ B\mapsto & {} t_3 t_0+(t_3 t_0)^{-1}, \\ C\mapsto & {} t_2 t_0+(t_2 t_0)^{-1}. \end{aligned}$$We classify the lattices of \(\triangle _q\)-submodules of finite-dimensional irreducible \({\mathfrak {H}}_q\)-modules. As a corollary, for any finite-dimensional irreducible \({\mathfrak {H}}_q\)-module V, the \(\triangle _q\)-module V is completely reducible if and only if \(t_0\) is diagonalizable on V.
中文翻译：
通用 Askey-Wilson 代数和 $$(C_1^\vee ,C_1)$$ ( C 1 ∨ , C 1 ) 类型的 DAHA 的有限维模
假设\({\mathbb {F}}\)是一个代数闭域，让q表示\({\mathbb {F}}\)中的一个非单位根的非零标量。通用 Askey–Wilson 代数\(\triangle _q\)是由生成器和关系定义的单位结合\({\mathbb {F}}\) -代数。生成器是A , B , C并且关系表明每个
$$\begin{aligned} A+\frac{q BC-q^{-1} CB}{q^2-q^{-2}}, \qquad B+\frac{q CA-q^{-1} AC}{q^2-q^{-2}}, \qquad C+\frac{q AB-q^{-1} BA}{q^2-q^{-2}} \end{aligned}$ $位于\(\triangle _q\) 的中心。类型为\((C_1^\vee ,C_1)\)的通用 DAHA（双仿射 Hecke 代数）\({\mathfrak {H}}_q \)是单位结合\({\mathbb {F}}\) -由\(\{t_i^{\pm 1}\}_{i=0}^3\)生成的代数，并且关系表明
$$\begin{aligned}&t_it_i^{-1}=t_i^{-1} t_i=1 \quad \hbox {for all } i=0,1,2,3;\\&t_i+t_i^{-1 } \hbox { 是中心} \quad \hbox {for all } i=0,1,2,3; \\&\qquad \qquad \qquad t_0t_1t_2t_3=q^{-1}。\end{对齐}$$每个\({\mathfrak {H}}_q\) -module 都是一个\(\triangle _q\) -module 通过注入\(\triangle _q\rightarrow {\mathfrak {H}}_q\)给定经过
$$\begin{aligned} A\mapsto & {} t_1 t_0+(t_1 t_0)^{-1}, \\ B\mapsto & {} t_3 t_0+(t_3 t_0)^{-1}, \\ C\mapsto & {} t_2 t_0+(t_2 t_0)^{-1}。\end{对齐}$$我们对有限维不可约\({\mathfrak {H}}_q\) -模块的\(\triangle _q\) -子模块的格进行分类。作为推论，对于任何有限维不可约\({\mathfrak {H}}_q\) -模V，\(\triangle _q\) -模V是完全可约的当且仅当\(t_0\)是在V 上可对角化。
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