Tridiagonal pairs of q-Racah type and the q-tetrahedron algebra

JOURNAL OF PURE AND APPLIED ALGEBRA

卷:225

Article

Terwilliger, Paul¹ 

[1] Univ Wisconsin, Dept Math, 480 Lincoln Dr, Madison, WI 53706 USA

关键词: Tridiagonal pair;    q-Tetrahedron algebra;    Double lowering map;    Spin model;    Distance-regular graph;    Spin Leonard pair;   

DOI  :  10.1016/j.jpaa.2020.106632

来源: Elsevier

【 摘 要 】

Let Fdenote a field, and let Vdenote a vector space over Fwith finite positive dimension. We consider an ordered pair of F-linear maps A : V. Vand A*: V. Vsuch that (i) each of A, A* is diagonalizable; (ii) there exists an ordering {Vi} di= 0of the eigenspaces of Asuch that A*Vi. Vi-1+ Vi+ Vi+1for 0 = i = d, where V-1= 0and Vd+1= 0; (iii) there exists an ordering {V* i} di= 0of the eigenspaces of A* such that AV* i. V* i-1+ V*i+ V* i+1for 0 = i = d, where V* -1= 0and V* d+1= 0; (iv) there does not exist a subspace Uof Vsuch that AU. U, A* U subset of U-,U- U = 0, U = V. We call such a pair a tridiagonal pair on V. We assume that A, A* belongs to a family of tridiagonal pairs said to have q-Racah type. There is an infinite-dimensional algebra qcalled the q-tetrahedron algebra; it is generated by four copies of Uq(sl2) that are related in a certain way. Using A, A* we construct two q-module structures on V. In this construction the two main ingredients are the double lowering map.: V. Vdue to Sarah Bockting-Conrad, and a certain invertible map W: V. Vmotivated by the spin model concept due to V. F. R. Jones. (C) 2020 Elsevier B.V. All rights reserved.

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