| 摘要: | 在我和老師的會議中，認為型態(C1V, C1)的通用加法DAHA中能看到Leonard三元組，所以我們利用學術網站上的相關論文，並且透過通用 Racah代數來得到以下結果。
假設 'IF' 是一個特徵為零的代數封閉體。通用Racah代數R是由A,B,C,D生成的單位結合 IF-代數，關係式為 [A, B] = [B, C] = [C, A] = 2D 並且每個
[A, D] + AC - BA, [B, D] + BA - CB, [C, D] + CB – AC
在R上皆可換。
型態 (C1V, C1) 的通用加法 DAHA (雙仿射 Hecke 代數)H是由 t_0,t_1,t_2,t_3 生成的單位結合 IF-代數，關係式為
t_0+t_1+t_2+t_3= -1，
t_0^2,
t_1^2,
t_2^2,
t_3^2 皆可換。
任何H-module 都可以被認為是一個R-module 透過 IF-代數同態將 送到H，由下式給出
A 送到 (t_0+t_1-1)(t_0+t_1+1)/4，
B 送到 (t_0+t_2-1)(t_0+t_2+1)/4，
C 送到 (t_0+t_3-1)(t_0+t_3+1)/4。
令 V 表示有限維不可分 H-module。 在本文中，我們展示了 A, B, C
在 V 上可對角化若且為若 A, B, C 在 R-module V 的所有合成因子上為Leonard 三元組。
;In the meeting, I thought that the Leonard triples can be seen in the universal additive DAHA of type (C1V, C1), so we used the relevant papers on the academic website and obtained the following results through the universal Racah algebra.
Suppose that 'IF' is an algebraically closed field with characteristic 0. The universal Racah algebra R is a unital associative IF-algebra generated by A, B, C, D and the relations state that [A, B] = [B, C] = [C, A] = 2D and each of
[A, D] + AC - BA, [B, D] + BA - CB, [C, D] + CB - AC
is central in R. The universal additive DAHA (double affine Hecke algebra) H of type (C1V, C1) is a unital associative IF-algebra generated by t_i (i=0,1,2,3) and the relations state that
t_0+t_1+t_2+t_3= -1，
t_i^2 is central for all i = 0, 1, 2, 3.
Any H-module can be considered as a R-module via the IF-algebra homomorphism R to H given by
A mapsto (t_0+t_1-1)(t_0+t_1+1)/4,
B mapsto (t_0+t_2-1)(t_0+t_2+1)/4,
C mapsto (t_0+t_3-1)(t_0+t_3+1)/4.
Let V be a finite-dimensional irreducible H-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 each composition factor of the R-module V. |
