Kauffman skein algebras and quantum Teichmüller spaces via factorization homology

Cooke, Juliet

doi:10.1142/s0218216520500893

Cited by 6 publications

(7 citation statements)

References 22 publications

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“…Algebras over three strands such as the Temperley-Lieb or Brauer ones that arise in Schur-Weyl duality have been shown to be quotients of Racah algebras [8,6]. Isomorphisms with certain Kauffman-Skein algebras have been established [4,5]. Howe duality could be used to relate different presentations [19,20,18].…”

Section: Introductionmentioning

confidence: 99%

Racah Problems for the Oscillator Algebra, the Lie Algebra $$\mathfrak {sl}_n$$, and Multivariate Krawtchouk Polynomials

Crampé

Vijver

Vinet

2020

Ann. Henri Poincaré

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The oscillator Racah algebra Rn(h) is realized by the intermediate Casimir operators arising in the multifold tensor product of the oscillator algebra h. An embedding of the Lie algebra sl n−1 into Rn(h) is presented. It relates the representation theory of the two algebras. We establish the connection between recoupling coefficients for h and matrix elements of slnrepresentations which are both expressed in terms of multivariate Krawtchouk polynomials of Griffiths type.

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Section: Introductionmentioning

confidence: 99%

Racah Problems for the Oscillator Algebra, the Lie Algebra $$\mathfrak {sl}_n$$, and Multivariate Krawtchouk Polynomials

Crampé

Vijver

Vinet

2020

Ann. Henri Poincaré

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“…We will use the isomorphism between the subalgebra L Uq(sl2) Σ0,n+1 of the Aleeksev moduli algebra L Σ0,n+1 which is invariant under the action of U q (sl 2 ) and the Askey-Wilson algebra AW(n) from Section 4, and also the isomorphism between L Uq(sl2) Σg,r and the skein algebra Sk q (Σ g,r ) from Section 5, so that we can instead compute the Hilbert series of L Uq(sl2) Σg,r whose compution is a generalisations of the calculations for Σ 0,4 by the first author in [Coo20].…”

Section: Graded Dimensionsmentioning

confidence: 99%

“…Theorem 1.1 is a generalisation of the classical result that AW(3) is isomorphic to the Kauffman bracket skein algebra of the four-punctured sphere. This was proven by showing that AW(3) is isomorphic to the (C ∨ 1 , C 1 ) spherical double affine Hecke algebra (DAHA) [Koo07;Ter13] and by comparing the presentation of the (C ∨ 1 , C 1 ) spherical DAHA to the presentation of the Kauffman bracket skein algebra of the fourpunctured sphere [BS18;Coo20;Hik19]. This approach is not readily generalisable so we will instead prove Theorem 1.1 by chaining together the following three maps Sk q (Σ 0,n+1 ) ⊆ Sk st q (Σ 0,n+1 ) ∼ − → O q (sl 2 ) ⊗n ∼ − → U q (sl 2 ) lf ⊗n − → U q (sl 2 ) ⊗n , which we shall now discuss.…”

Section: Introductionmentioning

confidence: 99%

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Higher Rank Askey-Wilson Algebras as Skein Algebras

Cooke¹,

Lacabanne²

2022

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In this paper we give a topological interpretation and diagrammatic calculus for the rank (n − 2) Askey-Wilson algebra by proving there is an explicit isomorphism with the Kauffman bracket skein algebra of the (n + 1)-punctured sphere. To do this we consider the Askey-Wilson algebra in the braided tensor product of n copies of either the quantum group Uq(sl 2 ) or the reflection equation algebra. We then use the isomorpism of the Kauffman bracket skein algebra of the (n + 1)-punctured sphere with the Uq(sl 2 ) invariants of the Aleeksev moduli algebra to complete the correspondence. We also find the graded vector space dimension of the Uq(sl 2 ) invariants of the Aleeksev moduli algebra and apply this to finding a presentation of the skein algebra of the five-punctured sphere and hence also find a presentation for the rank 2 Askey-Wilson algebra. Contents 1. Introduction 1 2. Askey-Wilson Algebras 5 3. Braided Tensor Product of copies of the locally finite part of U q (sl 2 ) 8 4. Moduli algebras 14 5. Skein Algebras 17 6. Commutator Relations 20 7. Action of the braid group 21 8. Graded Dimensions 23 9. Presentation of the Skein Algebra of the Five-Punctured Sphere 25 A. Appendix 34 References 37

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“…Subsequently, there has been an enormous amount of work at this crossroads between quantum field theory, representation theory, topology, and higher category theory. To give just a few examples, we refer to the research program of Ben-Zvi, Gunningham, Nadler and collaborators [BN09; BN18; BGN19], the work of Ben-Zvi, Brochier and Jordan [BBJ18a;BBJ18b] and the results on skein algebras that have followed them [Coo20;GJS19], and the work of Frenkel and Gaiotto [Gai18;FG20]. Much of the mathematical work has used methods involving higher categories and derived geometry, in much the same spirit as homological mirror symmetry reworks the physicists' view on duality for N = (2, 2) supersymmetric sigma models.…”

Section: Introductionmentioning

confidence: 99%

Higher Deformation Quantization for Kapustin-Witten Theories

Elliott¹,

Gwilliam²,

Williams³

2021

Preprint

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We pursue a uniform quantization of all twists of 4-dimensional N = 4 supersymmetric Yang-Mills theory, using the BV formalism, and we explore consequences for factorization algebras of observables. Our central result is the construction of a one-loop exact quantization on R 4 for all such twists and for every point in a moduli of vacua. When an action of the group SO(4) can be defined -for instance, for Kapustin and Witten's family of twists -the associated framing anomaly vanishes. It follows that the local observables in such theories can be canonically described by a family of framed E 4 algebras; this structure allows one to take the factorization homology of observables on any oriented 4-manifold. In this way, each Kapustin-Witten theory yields a fully extended, oriented 4-dimensional topological field theory à la Lurie and Scheimbauer. Contents 1. Introduction 2.1. Classical BV Formalism 2.2. Quantum BV formalism 2.3. Equivariant BV formalism 3. Twists of 4d N = 4 Super Yang-Mills Theory 3.1. Summary of Twists 3.2. Anomaly Vanishing 4. Perturbation Theory Around a Vacuum 4.1. Moduli of Vacua 4.2. Families of Classical Field Theories over [g * /G] 4.3. Families of Quantum Field Theories over [g * /G]

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Kauffman skein algebras and quantum Teichmüller spaces via factorization homology

Cited by 6 publications

References 22 publications

Racah Problems for the Oscillator Algebra, the Lie Algebra $$\mathfrak {sl}_n$$, and Multivariate Krawtchouk Polynomials

Racah Problems for the Oscillator Algebra, the Lie Algebra $$\mathfrak {sl}_n$$, and Multivariate Krawtchouk Polynomials

Higher Rank Askey-Wilson Algebras as Skein Algebras

Higher Deformation Quantization for Kapustin-Witten Theories

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