Linear Maps That Act Tridiagonally with Respect to Eigenbases of the Equitable Generators of U q ( sl 2 )

Hasan Alnajjar and Brian Curtin
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Hasan Alnajjar: Department of Mathematics, The University of Jordan, Amman 11942, Jordan
Brian Curtin: Department of Mathematics and Statistics, University of South Florida, 4202 E. Fowler Ave. CMC342, Tampa, FL 33620, USA

Mathematics, 2020, vol. 8, issue 9, 1-13

Abstract: 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 U q ( sl 2 ) over F . Let V denote a finite-dimensional irreducible U q ( sl 2 ) -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 , X Y , Y Z , and Z X on V give a basis for R when d ≥ 3 .

Keywords: finite-dimensional U q ( sl 2 )-modules; standard eigenbasis; Leonard pairs (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2020
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