Details

Autor(en) / Beteiligte

• Balmaceda, Jose Maria P.
• Maralit, Jryl P.

Titel

Leonard triples from Leonard pairs constructed from the standard basis of the Lie algebra sl2

Ist Teil von

• Linear algebra and its applications, 2012-10, Vol.437 (7), p.1961-1977

Ort / Verlag

Amsterdam: Elsevier Inc

Erscheinungsjahr

2012

Link zum Volltext

Quelle

Elektronische Zeitschriftenbibliothek - Frei zugängliche E-Journals

Beschreibungen/Notizen

• 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 Kd+1 if we can find invertible matrices S1 and S2∈Md+1(K) such that (i) S1-1AS1 is diagonal and S1-1A∗S1 is irreducible tridiagonal, and (ii) S2-1A∗S2 is diagonal and S2-1AS2 is irreducible tridiagonal. We extend this concept to three matrices. We say an ordered triple of matrices A,A∗ and Aϵ is a Leonard triple on Kd+1 if we can find invertible matrices S1, S2 and S3 such that (i) S1-1AS1 is diagonal and both S1-1A∗S1 and S1-1AϵS1 are irreducible tridiagonal, (ii) S2-1A∗S2 is diagonal and both S2-1AS2 and S2-1A∗S2 are irreducible tridiagonal and (iii) S3-1AϵS3 is diagonal and both S3-1AS3 and S3-1A∗S3 are irreducible tridiagonal. Let A=0d010d-120⋱⋱⋱10d0andA∗=diag(d,d-2,…,-d) be (d+1)×(d+1) matrices. Then A,A∗ is a Leonard pair on Kd+1. We determine all the matrices Aϵ such that A,A∗,Aϵ form a Leonard triple on Kd+1.