| 姓名 |
黃曦嶢(Si-Yao Huang)
查詢紙本館藏 |
畢業系所 |
數學系 |
| 論文名稱 |
(The Leonard triples and the universal additive DAHA of type (C1ˇ,C1))
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| 檔案 |
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[Bibtex 格式]
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| 摘要(中) |
在我和老師的會議中，認為型態(C1∨, 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 ? 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?-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 (C1∨, 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 ?-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 ?-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。 |
| 摘要(英) |
In the meeting, I thought that the Leonard triples can be seen in the universal additive DAHA of type (C1∨, C1), so we used the relevant papers on the academic website and obtained the following results through the universal Racah algebra.
Suppose that ? is an algebraically closed field with characteristic 0. The universal Racah algebra R is a unital associative ?-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 (C1∨, C1) is a unital associative ?-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 ?-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. |
| 關鍵字(中) |
★ 三元組 |
關鍵字(英) |
★ Leonard
★ DAHA |
| 論文目次 |
中文摘要------------------------------------------i
英文摘要----------------------------------------ii
誌謝-------------------------------------------iii
目錄--------------------------------------------iv
表目錄---------------------------------------------v
Introduction---------------------------------1
From the Racah polynomials to the universal Racah algebra-----------------------------------------4
Preliminaries on the irreducible R-modules with finite dimensions--------------------------------------6
The necessary and sufficient conditions for A,B,C as a Leonard triple on irreducible R-modules with finite dimensions--------------------------------------7
Preliminaries on the irreducible H-modules with even dimensions--------------------------------------9
The conditions for A,B,C as diagonalizable on irreducible H-modules with even dimensions------------------11
Preliminaries on the irreducible H-modules with odd dimensions--------------------------------------17
The conditions for A,B,C as diagonalizable on irreducible H-modules with odd dimensions-------------------19
References-------------------------------------23 |
| 參考文獻 |
1. B. Curtin, Modular Leonard triples, Linear Algebra and its Applications 424 (2007), 510–539.
2. V.X. Genest, L. Vinet, and A. Zhedanov, Embeddings of the Racah algebra into the Bannai–Ito
algebra, SIGMA 11 (2015), 050, 11 pp.
3. ____, The non-symmetric Wilson polynomials are the Bannai–Ito polynomials, Proceedings
of the American Mathematical Society 144 (2016), 5217–5226.
4. W. Groenevelt, Fourier transforms related to a root system of rank 1, Transformation Groups
12 (2007), 77–116.
5. H.-W. Huang, Finite-dimensional modules of the universal Racah algebra and the universal
additive DAHA of type (C1∨, C1), accepted by Journal of Pure and Applied Algebra.
6. ____, The universal DAHA of type (C1∨, C1) and Leonard triples, accepted by Communications
in Algebra.
7. ____, Finite-dimensional irreducible modules of the Bannai–Ito algebra at characteristic zero,
Letters in Mathematical Physics 110 (2020), 2519–2541.
8. ____, The Racah algebra as a subalgebra of the Bannai–Ito algebra, SIGMA 16 (2020), 075,
15 pages.
9. H.-W. Huang and S. Bockting-Conrad, Finite-dimensional irreducible modules of the Racah
algebra at characteristic zero, SIGMA 16 (2020), 018, 17 pages.
10. P. Terwilliger, Two linear transformations each tridiagonal with respect to an eigenbasis of the
other, Linear Algebra and Its Applications 330 (2001), 149–203. |
| 指導教授 |
黃皜文(Hao-Wen Huang)
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審核日期 |
2021-12-13 |
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