Some q-exponential Formulas for Finite-Dimensional $\square _{q}$-Modules

Yang, Yang

doi:10.1007/s10468-019-09862-y

Cited by 5 publications

(8 citation statements)

References 18 publications

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“…The finite-dimensional q -modules are investigated in [33,34]. By [34,Proposition 5.2], each q -generator is invertible on every nonzero finite-dimensional q -module.…”

Section: Introductionmentioning

confidence: 99%

“…The finite-dimensional q -modules are investigated in [33,34]. By [34,Proposition 5.2], each q -generator is invertible on every nonzero finite-dimensional q -module. This result gets used in [34,Sections 8 and 9] to obtain some remarkable q-exponential formulas.…”

Section: Introductionmentioning

confidence: 99%

“…By [34,Proposition 5.2], each q -generator is invertible on every nonzero finite-dimensional q -module. This result gets used in [34,Sections 8 and 9] to obtain some remarkable q-exponential formulas. In [33], the finite-dimensional irreducible q -modules are classified up to isomorphism.…”

Section: Introductionmentioning

confidence: 99%

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An Infinite-Dimensional q-Module Obtained from the q-Shuffle Algebra for Affine sl<sub>2</sub>

Post

Terwilliger

2020

SIGMA

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Let F denote a field, and pick a nonzero q ∈ F that is not a root of unity. Let Z 4 = Z/4Z denote the cyclic group of order 4. Define a unital associative F-algebra q by generators {x i } i∈Z4 and relationswhenever x 1 ξ = 0 and x 3 ξ = 0 and ξ = 0. The q -module V is called NIL whenever V is generated by a NIL vector. We show that up to isomorphism there exists a unique NIL qmodule, and it is irreducible and infinite-dimensional. We describe this module from sixteen points of view. In this description an important role is played by the q-shuffle algebra for affine sl 2 .

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“…The finite-dimensional q -modules are investigated in [33,34]. By [34,Proposition 5.2], each q -generator is invertible on every nonzero finite-dimensional q -module.…”

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

An Infinite-Dimensional q-Module Obtained from the q-Shuffle Algebra for Affine sl<sub>2</sub>

Post

Terwilliger

2020

SIGMA

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show abstract

“…The finite-dimensional q -modules are investigated in [33,34]. By [33,Proposition 5.2], each q -generator is invertible on every nonzero finite-dimensional q -module.…”

Section: Introductionmentioning

confidence: 99%

An Infinite-Dimensional $\square_q$-Module Obtained from the $q$-Shuffle Algebra for Affine $\mathfrak{sl}_2$

Post,

Terwilliger

2018

Preprint

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Let F denote a field, and pick a nonzero q ∈ F that is not a root of unity. Let Z 4 = Z/4Z denote the cyclic group of order 4. Define a unital associative F-algebra q by generators {x i } i∈Z 4 and relationswhere [3] q = (q 3 − q −3 )/(q − q −1 ). Let V denote a q -module. A vector ξ ∈ V is called NIL whenever x 1 ξ = 0 and x 3 ξ = 0 and ξ = 0. The q -module V is called NIL whenever V is generated by a NIL vector. We show that up to isomorphism there exists a unique NIL q -module, and it is irreducible and infinite-dimensional. We describe this module from sixteen points of view. In this description an important role is played by the q-shuffle algebra for affine sl 2 .

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“…The paper [121] shows howĤ q is related to Leonard pairs. Additional results in the literature link tridiagonal pairs and Leonard pairs with the Lie algebra sl 2 (see [3, 7-9, 26, 81, 120]), the quantum algebras U q (sl 2 ) (see [2,29,30,88,147,162]), U q ( sl 2 ) (see [5,24,41,75,76,84,152]), the tetrahedron Lie algebra (see [25,56,79,96]), and its q-deformation q (see [42,73,77,78,80,164,165]).…”

Section: Casementioning

confidence: 99%

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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Some q-exponential Formulas for Finite-Dimensional $\square _{q}$-Modules

Cited by 5 publications

References 18 publications

An Infinite-Dimensional q-Module Obtained from the q-Shuffle Algebra for Affine sl<sub>2</sub>

An Infinite-Dimensional q-Module Obtained from the q-Shuffle Algebra for Affine sl<sub>2</sub>

An Infinite-Dimensional $\square_q$-Module Obtained from the $q$-Shuffle Algebra for Affine $\mathfrak{sl}_2$

Totally bipartite tridiagonal pairs

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