论文标题
Askey-Wilson代数及其头像
The Askey-Wilson algebra and its avatars
论文作者
论文摘要
Zhedanov引入的原始Askey-Wilson代数编码同名多项式的双光谱属性。目前使用的名称“ Askey-Wilson代数”来指代各种相关结构，这些结构出现在许多上下文中。我们审查这些版本，对其进行整理并建立它们之间的关系。我们专注于两个特定的化身。第一个是原始Zhedanov代数的商；在$ d_4 $类型的Weyl组下，它是不变的，并具有反射代数介绍。第二个是第一个的普遍类似物。这是四函数球体的Kauffman支架绞线代数（KBSA）以及通用双仿射Hecke代数$（C_1^{\ VEE}，C_1）$的子代数。第二个代数来自$ u_q（\ mathfrak {sl} _2）$的RACAH问题，并通过其三倍张量产品中的$ u_q（\ Mathfrak {sl} _2）$与$ u_q（\ mathfrak {sl} _2）$相关。 Artin Braid小组如何通过$ r $ $ - 慕斯（在RACAH问题中）或半dehn Twist（在示意图的KBSA图片中）通过共轭（在RACAH问题中）进行连续轭的化身。简要讨论和总结了以图解的方式简要讨论和总结了定义更高排名的Askey-Wilson代数的尝试。
The original Askey-Wilson algebra introduced by Zhedanov encodes the bispectrality properties of the eponym polynomials. The name 'Askey-Wilson algebra' is currently used to refer to a variety of related structures that appear in a large number of contexts. We review these versions, sort them out and establish the relations between them. We focus on two specific avatars. The first is a quotient of the original Zhedanov algebra; it is shown to be invariant under the Weyl group of type $D_4$ and to have a reflection algebra presentation. The second is a universal analogue of the first one; it is isomorphic to the Kauffman bracket skein algebra (KBSA) of the four-punctured sphere and to a subalgebra of the universal double affine Hecke algebra $(C_1^{\vee},C_1)$. This second algebra emerges from the Racah problem of $U_q(\mathfrak{sl}_2)$ and is related via an injective homomorphism to the centralizer of $U_q(\mathfrak{sl}_2)$ in its threefold tensor product. How the Artin braid group acts on the incarnations of this second avatar through conjugation by $R$-matrices (in the Racah problem) or half Dehn twists (in the diagrammatic KBSA picture) is also highlighted. Attempts at defining higher rank Askey-Wilson algebras are briefly discussed and summarized in a diagrammatic fashion.
