Article ID Journal Published Year Pages File Type
4602472 Linear Algebra and its Applications 2009 23 Pages PDF
Abstract

The q-tetrahedron algebra ⊠q was recently introduced and has been studied in connection with tridiagonal pairs. In this paper we further develop this connection. Let K denote an algebraically closed field and let q denote a nonzero scalar in K that is not a root of unity. Let V denote a vector space over K with finite positive dimension and let A,A∗ denote a tridiagonal pair on V. Let , (resp. ) denote a standard ordering of the eigenvalues of A (resp. A∗). Ito and Terwilliger have shown that when θi=q2i-d and there exists an irreducible ⊠q-module structure on V such that the ⊠q generators x01,x23 act as A,A∗ respectively. In this paper we examine the case in which there exists a nonzero scalar c in K such that θi=q2i-d and for 0⩽i⩽d. In this case we associate to A,A∗ a polynomial P in one variable and prove the following theorem as our main result.Theorem The following are equivalent:1.[(i)] There exists a ⊠q-module structure on V such that x01 acts as A and x30+cx23 acts as A∗, where x01,x30,x23 are standard generators for ⊠q.2.[(ii)] P(q2d-2(q-q-1)-2)≠0.Suppose (i) and (ii) hold. Then the ⊠q-module structure on V is unique and irreducible.

Related Topics
Physical Sciences and Engineering Mathematics Algebra and Number Theory