Leonard triples from Leonard pairs constructed from the standard basis of the Lie algebra sl _2

摘要

Let K denote an algebraically closed field of characteristic zero and d ≥ 3 denote an integer. An ordered pair of matrices A, A is a Leonard pair on the vector space K ~(d+1) if we can find invertible matrices S _1 and S _2 ≡ M _(d+1)(K) such that (i) S _1 ~(-1)AS _1 is diagonal and S _1 ~(-1)AS _1 is irreducible tridiagonal, and (ii) S _2 ~(-1)AS _2 is diagonal and S _2 ~(-1)AS _2 is irreducible tridiagonal. We extend this concept to three matrices. We say an ordered triple of matrices AA and A ≡ is a Leonard triple on K d+1 if we can find invertible matrices S _1, S _2 and S _3 such that (i) S _1 ~(-1)AS _1 is diagonal and both S _1 ~(-1)AS _1 and S _1 ~(-1)A ≡S 1are irreducible tridiagonal, (ii) S _2 ~(-1)AS _2 is diagonal and both S _2 ~(-1)AS _2 and S _2 ~(-1) AS _2 are irreducible tridiagonal and (iii) S _3 ~(-1)A ≡S _3 is diagonal and both S ~(-1)AS _3 and S _3 ~(-1)AS _3 are irreducible tridiagonal. Let A=0d010d-12010d0and ~A=diag(d,d-2,...,-d) be (d +1) x (d +1) matrices. Then A, A is a Leonard pair on K ~(d+1).We determine all the matrices A ~≡ such that A, A, A ~≡ form a Leonard triple on K ~(d+1).