The Classification of Finite-Dimensional Irreducible Modules of Bannai/Ito Algebra

Hou, Bo; Wang, Meng; Gao, Suogang

doi:10.1080/00927872.2014.990030

Cited by 8 publications

(11 citation statements)

References 16 publications

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“…The idea of our classification comes from [15]. We mention that a similar issue arises in the case of the Bannai-Ito algebra BI [11] which is addressed by the first author in [12]. The result [14, Theorem 5.4] reveals that the Racah algebra ℜ is isomorphic to an F-subalgebra of BI.…”

Section: Introductionmentioning

confidence: 99%

Finite-Dimensional Irreducible Modules of the Racah Algebra at Characteristic Zero

Huang¹,

Bockting-Conrad

2020

SIGMA

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Assume that F is an algebraically closed field with characteristic zero. The Racah algebra ℜ is the unital associative F-algebra defined by generators and relations in the following way. The generators are A, B, C, D and the relations assert thatand that each ofIn this paper we discuss the finite-dimensional irreducible ℜ-modules in detail and classify them up to isomorphism. To do this, we apply an infinite-dimensional ℜ-module and its universal property. We additionally give the necessary and sufficient conditions for A, B, C to be diagonalizable on finite-dimensional irreducible ℜ-modules.

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Section: Introductionmentioning

confidence: 99%

Finite-Dimensional Irreducible Modules of the Racah Algebra at Characteristic Zero

Huang¹,

Bockting-Conrad

2020

SIGMA

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“…The Leonard triples are classified up to isomorphism in [67] (for q-Racah type); [45] (for Racah type); [89] (for Krawtchouk type); [65] (for Bannai/Ito type with even diameter); [63] (for Bannai/Ito type with odd diameter). Additional results on Leonard triples can be found in [60,92,99,150,155] (for q-Racah type); [9,94,125] (for Racah type); [14,95] (for Krawtchouk type); [27,32,47,48,62,66,156] (for Bannai/Ito type); [102] (for general case).…”

Section: Casementioning

confidence: 96%

Totally bipartite tridiagonal pairs

Nomura

Terwilliger²

2021

ELA

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There is a concept in linear algebra called a tridiagonal pair. The concept was motivated by the theory of $Q$-polynomial distance-regular graphs. We give a tutorial introduction to tridiagonal pairs, working with a special case as a concrete example. The special case is called totally bipartite, or totally bipartite (TB). Starting from first principles, we give an elementary but comprehensive account of TB tridiagonal pairs. The following topics are discussed: (i) the notion of a TB tridiagonal system; (ii) the eigenvalue array; (iii) the standard basis and matrix representations; (iv) the intersection numbers; (v) the Askey--Wilson relations; (vi) a recurrence involving the eigenvalue array; (vii) the classification of TB tridiagonal systems; (viii) self-dual TB tridiagonal pairs and systems; (ix) the $\mathbb{Z}_3$-symmetric Askey--Wilson relations; (x) some automorphisms and antiautomorphisms associated with a TB tridiagonal pair; and (xi) an action of the modular group ${\rm PSL}_2(\mathbb{Z})$ associated with a TB tridiagonal pair.

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“…Γ q A can be obtained from this operator upon repeatedly adding ⊗1 at the back. As ∆(1) = τ (1) = 1 ⊗ 1 and as α k cannot be ∆ ⊗ 1 by (25), this is equivalent to writing:…”

Section: 2mentioning

confidence: 99%

“…Finally, we may also apply the extension procedure (24)- (25) to the osp q (1|2)generators. The only sets that lend themselves to this extension are the sets [i; j] of consecutive integers.…”

Section: Ordered By Inclusion Such That the Operators γ Qmentioning

confidence: 99%

“…It was first introduced in [41] to encode the bispectral structure of the Bannai-Ito orthogonal polynomials, which were used in [2] for the classification of association schemes. Finite-dimensional irreducible representations of this algebra have been classified [25] and there is a deep connection with Leonard pairs [7]. The Bannai-Ito algebra is moreover isomorphic to a degeneration of the double affine Hecke algebra of type (C ∨ 1 , C 1 ), see [21].…”

Section: Introductionmentioning

confidence: 99%

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The Higher Rank q-Deformed Bannai-Ito and Askey-Wilson Algebra

Bie

Clercq

Vijver

2019

Commun. Math. Phys.

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The q-deformed Bannai-Ito algebra was recently constructed in the threefold tensor product of the quantum superalgebra osp q (1|2). It turned out to be isomorphic to the Askey-Wilson algebra. In the present paper these results will be extended to higher rank. The rank n − 2 q-Bannai-Ito algebra A q n , which by the established isomorphism also yields a higher rank version of the Askey-Wilson algebra, is constructed in the n-fold tensor product of osp q (1|2). An explicit realization in terms of q-shift operators and reflections is proposed, which will be called the Z n 2 q-Dirac-Dunkl model. The algebra A q n is shown to arise as the symmetry algebra of the constructed Z n 2 q-Dirac-Dunkl operator and to act irreducibly on modules of its polynomial null-solutions. An explicit basis for these modules is obtained using a q-deformed CK-extension and Fischer decomposition.

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The Classification of Finite-Dimensional Irreducible Modules of Bannai/Ito Algebra

Cited by 8 publications

References 16 publications

Finite-Dimensional Irreducible Modules of the Racah Algebra at Characteristic Zero

Finite-Dimensional Irreducible Modules of the Racah Algebra at Characteristic Zero

Totally bipartite tridiagonal pairs

The Higher Rank q-Deformed Bannai-Ito and Askey-Wilson Algebra

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