The Lusztig automorphism of Uq(𝔰𝔩2) from the equitable point of view

Terwilliger, Paul

doi:10.1142/s0219498817502358

Cited by 9 publications

(4 citation statements)

References 27 publications

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“…In this section we recall the algebra U q (sl 2 ) in its equitable presentation. For more information on this presentation, see [20,[30][31][32].…”

Section: Equitable Triplesmentioning

confidence: 99%

Tridiagonal pairs of $q$-Racah type and the $q$-tetrahedron algebra

Terwilliger¹

2020

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Let F denote a field, and let V denote a vector space over F with finite positive dimension. We consider an ordered pair of F-linear maps A :, where V * −1 = 0 and V * δ+1 = 0; (iv) there does not exist a subspace U of V such that AU ⊆ U , A * U ⊆ U , U = 0, U = V . We call such a pair a tridiagonal pair on V . We assume that A, A * belongs to a family of tridiagonal pairs said to have q-Racah type. There is an infinite-dimensional algebra ⊠ q called the q-tetrahedron algebra; it is generated by four copies of U q (sl 2 ) that are related in a certain way. Using A, A * we construct two ⊠ q -module structures on V . In this construction the two main ingredients are the double lowering map ψ : V → V due to Sarah Bockting-Conrad, and a certain invertible map W : V → V motivated by the spin model concept due to V. F. R. Jones.

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“…In this section we recall the algebra U q (sl 2 ) in its equitable presentation. For more information on this presentation, see [20,[30][31][32].…”

Section: Equitable Triplesmentioning

confidence: 99%

Tridiagonal pairs of $q$-Racah type and the $q$-tetrahedron algebra

Terwilliger¹

2020

Preprint

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“…It will be seen that the tridiagonalization of the q-difference operator, of which the little q-Jacobi polynomials are eigenfunctions, leads to the Askey-Wilson algebra in the fashion described above. It will be further observed that if the equitable presentation of U q (sl(2)) is called upon [ITW06,T15], the little q-Jacobi operator and its tridiagonalized companion turn out to naturally take the form that generators of the Askey-Wilson algebra have when symmetrically embedded in U q (sl(2)).…”

Section: Introductionmentioning

confidence: 99%

Little and big q-Jacobi polynomials and the Askey–Wilson algebra

et al. 2019

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The little and big q-Jacobi polynomials are shown to arise as basis vectors for representations of the Askey-Wilson algebra. The operators that these polynomials respectively diagonalize are identified within the Askey-Wilson algebra generated by twisted primitive elements of Uq(sl (2)). The little q-Jacobi operator and a tridiagonalization of it are shown to realize the equitable embedding of the Askey-Wilson algebra into Uq(sl(2)).MSC: 81R50; 81R10; 81U15; 39A70; 33D50; 39A13.

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“…q(1 − xy) q − q −1 . On every nonzero finite-dimensional U q (sl 2 )-module, x, y, z are invertible (see [18,Lemma 5.15]) and n x , n y , n z are nilpotent (see [18,Lemma 5.14]). Recall from [13, p. 204] the q-exponential function…”

Section: Introductionmentioning

confidence: 99%

“…Near the equation (1.1) we gave 18 equations for U q (sl 2 ). These equations were used to construct a rotator for U q (sl 2 ) (see [18,Definition 9.5]). These equations were also used to describe the Lusztig operators (see [11,12]) for U q (sl 2 ) in the equitable presentation (see [18,Theorem 9.9]).…”

Section: Introductionmentioning

confidence: 99%

Some $q$-exponential formulas for finite-dimensional $\square_q$-modules

Yang

2016

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We consider the algebra q which is a mild generalization of the quantum algebra Uq(sl 2 ). The algebra q is defined by generators and relations. The generators are {x i } i∈Z 4 , where Z 4 is the cyclic group of order 4. For i ∈ Z 4 the generators x i ,x i+1 satisfy a q-Weyl relation, and x i ,x i+2 satisfy a cubic q-Serre relation. For i ∈ Z 4 we show that the action of x i is invertible on every nonzero finite-dimensional q -module. We view x −1 i as an operator that acts on nonzero finite-dimensional q -modules. For i ∈ Z 4 , define n i,i+1 = q(1 − x i x i+1 )/(q − q −1 ). We show that the action of n i,i+1 is nilpotent on every nonzero finite-dimensional q -module. We view the q-exponential exp q (n i,i+1 ) as an operator that acts on nonzero finitedimensional q -modules. In our main results, for i, j ∈ Z 4 we express each of exp q (n i,i+1 )x j exp q (n i,i+1 ) −1 and exp q (n i,i+1 ) −1 x j exp q (n i,i+1 ) as a polynomial in {x ±1 k } k∈Z 4 .

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The Lusztig automorphism of Uq(𝔰𝔩2) from the equitable point of view

Abstract: We consider the quantum algebra U q (sl 2 ) in the equitable presentation. From this point of view, we describe the Lusztig automorphism and the corresponding Lusztig operator.

Cited by 9 publications

References 27 publications

Tridiagonal pairs of $q$-Racah type and the $q$-tetrahedron algebra

Tridiagonal pairs of $q$-Racah type and the $q$-tetrahedron algebra

Little and big q-Jacobi polynomials and the Askey–Wilson algebra

Some $q$-exponential formulas for finite-dimensional $\square_q$-modules

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