论文标题
$（C_1^\ vee，C_1）$和Leonard Triples的通用Daha
The universal DAHA of type $(C_1^\vee,C_1)$ and Leonard triples
论文作者
论文摘要
假设$ \ mathbb f $是一个代数封闭的字段，而$ q $是$ \ mathbb f $的非零标量，这不是统一的根源。通用的ASKEY - WILSON代数$ \ triangle_q $是由$ a，b，c $生成的一个Unital关联$ \ Mathbb f $ -Algebra，以及关系状态，$ a+a+a+a+\ frac {q bc-q bc-q q^{ - 1} cb} cb} cb} cb} cb} ca-q^{ - 1} ac} {q^2-q^{ - 2}}，\ qquad c+\ frac {q ab-q^{ - 1} ba} {q^2-q^{ - 2}} $ \ triangle_q $。类型$（C_1^\ vee，C_1）$的通用DAHA $ \ MATHFRAK H_Q $是由$ \ {t_i^{\ pm 1} \ pm 1} \} _ {i = 0}^3 $ and \ oft \ oft \ oft \ oft \ onge oggr of开始的Unital associative $ \ mathbb f $ -Algebra。 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 $}; \\ T_0T_1T_2T_3 = Q^{ - 1}。 \ end {caple*}给出了一个$ \ mathbb f $ -algebra同构$ \ triangle_q \ to \ mathfrak h_q $，它发送\ begin ob begin {eqnarray*} a＆\ mapsto＆\ mapsto＆t_1 t_1 t_1 t_0+（t_1 t_1 t_1 t_1 t_1 t_0） T_0+（T_3 T_0）^{ - 1}，\\ c＆\ mapsto＆t_2 T_0+（T_2 T_0）^{ - 1}。 \ end {eqnarray*}因此，任何$ \ mathfrak h_q $ -module都可以被视为$ \ triangle_q $ -module。令$ v $表示有限维度不可约的$ \ Mathfrak H_Q $ -MODULE。在本文中，我们表明，$ a，b，c $在$ v $上是可对角线的，并且仅当$ a，b，c $作为$ \ triangle_q $ -module $ v $的所有组合因子上的伦纳德三倍。
Assume that $\mathbb F$ is an algebraically closed field and $q$ is 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 generated by $A,B, C$ 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 $\mathfrak H_q$ 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*} It was given an $\mathbb F$-algebra homomorphism $\triangle_q\to \mathfrak H_q$ that sends \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*} Therefore any $\mathfrak H_q$-module can be considered as a $\triangle_q$-module. Let $V$ denote a finite-dimensional irreducible $\mathfrak H_q$-module. In this paper we show that $A,B,C$ are diagonalizable on $V$ if and only if $A,B,C$ act as Leonard triples on all composition factors of the $\triangle_q$-module $V$.
