论文标题
通用ASKEY的有限维模块 - Wilson代数和Daha类型$（C_1^\ VEE，C_1）$
Finite-dimensional modules of the universal Askey--Wilson algebra and DAHA of type $(C_1^\vee,C_1)$
论文作者
论文摘要
假设$ \ mathbb f $是一个代数封闭的字段，让$ q $表示不是unity根部的$ \ mathbb f $中的非零标量。通用的ASKEY - WILSON代数$ \ triangle_q $是一个Unital关联$ \ Mathbb f $ - 由发电机和关系定义的algebra。生成器是$ a，b，c $和关系状态，$$ a+\ frac {q bc-q^{ - 1} cb} {q^2-q^{ - 2}}，\ qquad b+\ frac {q quad b+\ frac {q ca-q^ca-q^{ - 1} ac} ac} ac} ac} ac} c+\ \ frac {q ab-q^{ - 1} ba} {q^2-q^{ - 2}} $$在$ \ triangle_q $中是中心。通用DAHA（双仿射Hecke代数）$ \ type $的Mathfrak H_Q $（C_1^\ VEE，C_1）$是由$ \ \ \ \ \ \ \ {t_i^{\ pm 1}} nonand}的Unital associative $ \ mathbb f $ -Algebra，由$ \ \ \ \ {t_i^{t_i^{ t_it_i^{ - 1} = t_i^{ - 1} t_i = 1 \ quad \ hbox {对于所有$ i = 0,1,1,2,3 $}; \\ \ hbox {$ t_i+t_i^{ - 1} $ is central} \ quad \ hbox {对于所有$ i = 0,1,1,2,3 $}; \\ T_0T_1T_2T_3 = Q^{ - 1}。 \end{gather*} Each $\mathfrak H_q$-module is a $\triangle_q$-module by pulling back via the injection $\triangle_q\to \mathfrak H_q$ given by \begin{eqnarray*} 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_2 t_0+（t_2 t_0）^{ - 1}。 \ end {eqnarray*}我们对有限维差异$ \ mathfrak h_q $ -modules的$ \ triangle_q $ -submodules的晶格进行了分类。结果，对于任何有限的不可约的$ \ mathfrak h_q $ -module $ v $，$ \ triangle_q $ -module $ v $完全还原时，仅当$ t_0 $是$ v $上的二分法时，才完全还原。
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 $$ A+\frac{q BC-q^{-1} CB}{q^2-q^{-2}}, \qquad B+\frac{q CA-q^{-1} AC}{q^2-q^{-2}}, \qquad C+\frac{q AB-q^{-1} BA}{q^2-q^{-2}} $$ 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{gather*} t_it_i^{-1}=t_i^{-1} t_i=1 \quad \hbox{for all $i=0,1,2,3$}; \\ \hbox{$t_i+t_i^{-1}$ is central} \quad \hbox{for all $i=0,1,2,3$}; \\ t_0t_1t_2t_3=q^{-1}. \end{gather*} Each $\mathfrak H_q$-module is a $\triangle_q$-module by pulling back via the injection $\triangle_q\to \mathfrak H_q$ given by \begin{eqnarray*} 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{eqnarray*} We classify the lattices of $\triangle_q$-submodules of finite-dimensional irreducible $\mathfrak H_q$-modules. As a consequence, 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 $V$.
