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Linear Maps That Act Tridiagonally with Respect to Eigenbases of the Equitable Generators of Uq(sl2)

by 1,*,† and 2,†
1
Department of Mathematics, The University of Jordan, Amman 11942, Jordan
2
Department of Mathematics and Statistics, University of South Florida, 4202 E. Fowler Ave. CMC342, Tampa, FL 33620, USA
*
Author to whom correspondence should be addressed.
These authors contributed equally to this work.
Mathematics 2020, 8(9), 1546; https://doi.org/10.3390/math8091546
Received: 4 August 2020 / Revised: 22 August 2020 / Accepted: 4 September 2020 / Published: 10 September 2020
(This article belongs to the Section Algebra and Geometry)
Let F denote an algebraically closed field; let q be a nonzero scalar in F such that q is not a root of unity; let d be a nonnegative integer; and let X, Y, Z be the equitable generators of Uq(sl2) over F. Let V denote a finite-dimensional irreducible Uq(sl2)-module with dimension d+1, and let R denote the set of all linear maps from V to itself that act tridiagonally on the standard ordering of the eigenbases for each of X, Y, and Z. We show that R has dimension at most seven. Indeed, we show that the actions of 1, X, Y, Z, XY, YZ, and ZX on V give a basis for R when d3. View Full-Text
Keywords: finite-dimensional Uq(sl2)-modules; standard eigenbasis; Leonard pairs finite-dimensional Uq(sl2)-modules; standard eigenbasis; Leonard pairs
MDPI and ACS Style

Alnajjar, H.; Curtin, B. Linear Maps That Act Tridiagonally with Respect to Eigenbases of the Equitable Generators of Uq(sl2). Mathematics 2020, 8, 1546. https://doi.org/10.3390/math8091546

AMA Style

Alnajjar H, Curtin B. Linear Maps That Act Tridiagonally with Respect to Eigenbases of the Equitable Generators of Uq(sl2). Mathematics. 2020; 8(9):1546. https://doi.org/10.3390/math8091546

Chicago/Turabian Style

Alnajjar, Hasan; Curtin, Brian. 2020. "Linear Maps That Act Tridiagonally with Respect to Eigenbases of the Equitable Generators of Uq(sl2)" Mathematics 8, no. 9: 1546. https://doi.org/10.3390/math8091546

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