Mixed quantum skew Howe duality and link invariants of type A

Queffelec, Hoel; Sartori, Antonio

doi:10.48550/arxiv.1504.01225

Cited by 8 publications

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“…Example A.5 ((3.15) of [QS19]). For the vector representation V of U q (gl(1|1)), the super quantum exterior algebra ∧ * q V is the quotient of the tensor (super)algebra T * (V ) by the ideal generated by the elements |0 ⊗ |0 and |0 ⊗ |1 + q −1 |1 ⊗ |0 of T 2 (V ) = V ⊗ V .…”

Section: Change-of-basis Bimodulesmentioning

confidence: 99%

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Trivalent vertices and bordered knot Floer homology in the standard basis

Manion¹

2020

Preprint

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We define new algebras, local bimodules, and bimodule maps in the spirit of Ozsváth-Szabó's bordered knot Floer homology. We equip them with the structure of 2representations of the categorified negative half U − of U q (gl(1|1)), 1-morphisms of such, and 2-morphisms respectively, and show that they categorify representations of U q (gl(1|1) − ) and maps between them. Unlike with Ozsváth-Szabó's algebras, the algebras considered here can be built from a higher tensor product operation recently introduced by Rouquier and the author.Our bimodules are all motivated by holomorphic disk counts in Heegaard diagrams; for positive and negative crossings, the bimodules can also be expressed as mapping cones involving a singular-crossing bimodule and the identity bimodule. In fact, they arise from an action of the monoidal category of Soergel bimodules via Rouquier complexes in the usual way, the first time (to the author's knowledge) such an expression has been obtained for braiding bimodules in Heegaard Floer homology.Furthermore, the singular crossing bimodule naturally factors into two bimodules for trivalent vertices; such bimodules have not appeared in previous bordered-Floer approaches to knot Floer homology. The action of the Soergel category comes from an action of categorified quantum gl(2) on the 2-representation 2-category of U − in line with the ideas of skew Howe duality, where the trivalent vertex bimodules are associated to 1-morphisms E, F in categorified quantum gl(2).

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Section: Change-of-basis Bimodulesmentioning

confidence: 99%

“…Remark A.2. We follow the conventions of Queffelec-Sartori [QS19], which agree (as far as we are aware) with those in [Sart15,Sart16].…”

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confidence: 99%

Trivalent vertices and bordered knot Floer homology in the standard basis

Manion¹

2020

Preprint

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“…) Note that we could have considered a slightly more general situation, allowing the quantum group on the right-hand side above to also be super (see [28,Theorem 4.2…”

Section: Skew-howe Dualitymentioning

confidence: 99%

“…We can now come to the statement of quantum skew Howe duality, referring to [17,2,5,7,39,28] for more details: Theorem 29. In the following situation:…”

Section: And Proposition 43]) the Above Version Appears In [39 Theore...mentioning

confidence: 99%

“…These issues motivated the introduction with Antonio Sartori of the notion of doubled Schur algebra [28], which is based on skew Howe duality with duals, and is closely related to the Brauer category at generic parameter β, thus giving a quantum description of the HOMFLY-PT polynomial. This approach was later categorified by Naisse and Vaz [26].…”

Section: Example 35 a Very Easy Example But One I Actually Quite Like...mentioning

confidence: 99%

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Polynomial link invariants and quantum algebras

Queffelec¹

2021

Preprint

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The definition of the Jones polynomial in the 80's gave rise to a large family of so-called quantum link invariants, based on quantum groups. These quantum invariants are all controlled by the same two-variable invariant (the HOMFLY-PT polynomial), which also specializes to the older Alexander polynomial. Building upon quantum Schur-Weyl duality and variants of this phenomenon, I will explain an algebraic setup that allows for global definitions of these quantum polynomials, and discuss extensions of these quantum objects designed to encompass all of the mentioned invariants, including the HOMFLY-PT polynomial.

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The first and second fundamental theorems of invariant theory for the quantum general linear supergroup

Zhang

2020

Journal of Pure and Applied Algebra

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We develop the non-commutative polynomial version of the invariant theory for the quantum general linear supergroup U q pgl m|n q. A non-commutative U q pgl m|n q-module superalgebra P k|l r|s is constructed, which is the quantum analogue of the supersymmetric algebra over C k|l b C m|n ' C r|s b pC m|n q˚. We analyse the structure of the subalgebra of U q pgl m|n q-invariants in P k|l r|s by using the quantum super analogue of Howe duality.The subalgebra of U q pgl m|n q-invariants in P k|l r|s is shown to be finitely generated. We determine its generators and establish a surjective superalgebra homomorphism from a braided supersymmetric algebra onto it. This establishes the first fundamental theorem of invariant theory for U q pgl m|n q.We show that the above mentioned superalgebra homomorphism is an isomorphism if and only if m ě mintk, ru and n ě mintl, su, and obtain a PBW basis for the subalgebra of invariants in this case. When the homomorphism is not injective, we give a representation theoretical description of the generating elements of the kernel. This way we obtain the relations obeyed by the generators of the subalgebra of invariants, producing the second fundamental theorem of invariant theory for U q pgl m|n q.We consider the special case with n " 0 in greater detail, obtaining a complete treatment of the non-commutative polynomial version of the invariant theory for U q pgl m q. In particular, the explicit SFT proved here is believed to be new. We also recover the FFT and SFT of the invariant theory for the general linear superalgebra from the classical limit (i.e., q Ñ 1) of our results.

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Mixed quantum skew Howe duality and link invariants of type A

Cited by 8 publications

References 0 publications

Trivalent vertices and bordered knot Floer homology in the standard basis

Trivalent vertices and bordered knot Floer homology in the standard basis

Polynomial link invariants and quantum algebras

The first and second fundamental theorems of invariant theory for the quantum general linear supergroup

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