Leonard pairs from the equitable basis of sl2

Alnajjar, Hasan; Curtin, Brian

doi:10.13001/1081-3810.1389

Cited by 21 publications

(23 citation statements)

References 11 publications

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“…We show that the converse is also true. (1). The coefficients of x s , y s , and z s are zero by (2) …”

Section: Leonard Pairs On Irreducible Sl 2 -Modulesmentioning

confidence: 99%

“…The nicest of Lie algebras is sl 2 , and an associated Leonard pair of Krawtchouk type was described in [26]. In [1], the authors described when two special elements of sl 2 act as a Leonard pair on an irreducible sl 2 -module. It was noted that the Leonard pairs which arise from that construction may be of Racah, Hahn, dual Hahn, or Krawtchouk type.…”

Section: Introductionmentioning

confidence: 99%

“…In this paper we describe when the Leonard pairs constructed from sl 2 in [1] can extend to a Leonard triple. Our work is similar to both [2] which take a different approach to the topic, and [11] which treats Leonard triples related to U q (sl 2 ).…”

Section: Introductionmentioning

confidence: 99%

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Leonard triples from the equitable basis of sl2

Alnajjar

Curtin

2015

Linear Algebra and its Applications

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“…We show that the converse is also true. (1). The coefficients of x s , y s , and z s are zero by (2) …”

Section: Leonard Pairs On Irreducible Sl 2 -Modulesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Leonard triples from the equitable basis of sl2

Alnajjar

Curtin

2015

Linear Algebra and its Applications

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“…For more information on the tetrahedron algebra see [3,5,9,16]. For more information on the equitable basis for sl 2 see [1,4].The equitable presentation of U q (sl 2 ) was first introduced in [20]. This presentation was used in the study of the q-tetrahedron algebra [14].…”

mentioning

confidence: 99%

“…For more information on the tetrahedron algebra see [3,5,9,16]. For more information on the equitable basis for sl 2 see [1,4].…”

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confidence: 99%

Bidiagonal triples

Funk-Neubauer

2017

Linear Algebra and its Applications

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We introduce a linear algebraic object called a bidiagonal triple. A bidiagonal triple consists of three diagonalizable linear transformations on a finite-dimensional vector space, each of which acts in a bidiagonal fashion on the eigenspaces of the other two. The concept of bidiagonal triple is a generalization of the previously studied and similarly defined concept of bidiagonal pair. We show that every bidiagonal pair extends to a bidiagonal triple, and we describe the sense in which this extension is unique. In addition we generalize a number of theorems about bidiagonal pairs to the case of bidiagonal triples. In particular we use the concept of a parameter array to classify bidiagonal triples up to isomorphism. We also describe the close relationship between bidiagonal triples and the representation theory of the Lie algebra sl 2 and the quantum algebra U q (sl 2 ).of parameter array to classify bidiagonal triples up to isomorphism (see Theorem 4.3). The statement of this classification does not make it clear how to construct the bidiagonal triples in each isomorphism class. Hence, we show how to construct bidiagonal triples using finitedimensional modules for the Lie algebra sl 2 and the quantum algebra U q (sl 2 ) (see Theorem 4.8). The finite-dimensional modules for sl 2 and U q (sl 2 ) are well known (see Lemmas 3.5 and 3.12).Bidiagonal triples and pairs originally arose in the study of the well-known quantum algebras U q ( sl 2 ) and U q (sl 2 ). The discovery of the so called equitable presentations of these algebras was the initial motivation for defining a bidiagonal triple. These equitable presentations were discovered during the attempt to classify a linear algebraic object called a tridiagonal pair. See below for more information on the equitable presentations and tridiagonal pairs. Thus, the importance of bidiagonal triples lies in the fact that they provide insight into the relationships between several closely connected algebraic objects. Although this paper is the first to explicitly define bidiagonal triples, they appear implicitly in [2,6,7,15,20,26]. In [2] bidiagonal triples were involved in constructing irreducible U q ( sl 2 )-modules from the Borel subalgebra of U q ( sl 2 ). In [6] bidiagonal triples were used in constructing U q ( sl 2 )-modules from certain raising and lowering maps satisfying the q-Serre relations. In [7] bidiagonal triples were present in using a certain type of tridiagonal pair to construct irreducible modules for the q-tetrahedron algebra. See below for more information on the q-tetrahedron algebra. In [15] bidiagonal triples were also present in using a certain type of tridiagonal pair to construct irreducible U q ( sl 2 )-modules. Bidiagonal triples were used in [20] to show that the generators from the equitable presentation of U q (sl 2 ) have invertible actions on each finite-dimensional U q (sl 2 )-module. See [26, Theorem 18.3] for another example of how bidiagonal triples appear in the representation theory of U q (sl 2 ).We now discuss the equitable ba...

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