Tensor product categorifications, Verma modules and the blob 2-category

Lacabanne, Abel; Naisse, Grégoire; Vaz, Pedro

doi:10.4171/qt/156

Cited by 6 publications

(8 citation statements)

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“…It is worth mentioning that an analogue of Proposition 1 for α ∈ N was recently proven in [60] using abstract categorical and diagrammatic methods, while the case of α being a formal parameter was further treated in [61]. To our knowledge, Proposition 1 is the first result of this kind that treats q and α as actual complex numbers in some explicit allowed domains.…”

Section: The One-boundary Spin Chain and The Blob Algebramentioning

confidence: 90%

$U_q\mathfrak{sl}_2$-invariant non-compact boundary conditions for the XXZ spin chain

Chernyak¹,

Gainutdinov²,

Saleur³

2022

Preprint

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We introduce new Uqsl2-invariant boundary conditions for the open XXZ spin chain. For generic values of q we couple the bulk Hamiltonian to an infinite-dimensional Verma module on one or both boundaries of the spin chain, and for q = e iπ p a 2p-th root of unity -to its p-dimensional analogue. Both cases are parametrised by a continuous "spin" α ∈ C.To motivate our construction, we first specialise to q = i, where we obtain a modified XX Hamiltonian with unrolled quantum group symmetry, whose spectrum and scaling limit is computed explicitly using free fermions. In the continuum, this model is identified with the (η, ξ) ghost CFT on the upper-half plane with a continuum of conformally invariant boundary conditions on the real axis. The different sectors of the Hamiltonian are identified with irreducible Virasoro representations.Going back to generic q we investigate the algebraic properties of the underlying lattice algebras. We show that if q α / ∈ ±q Z , the new boundary coupling provides a faithful representation of the blob algebra which is Schur-Weyl dual to Uqsl2. Then, modifying the boundary conditions on both the left and the right, we obtain a representation of the universal two-boundary Temperley-Lieb algebra. The generators and parameters of these representations are computed explicitly in terms of q and α. Finally, we conjecture the general form of the Schur-Weyl duality in this case.This paper is the first in a series where we will study, at all values of the parameters, the spectrum and its continuum limit, the representation content of the relevant lattice algebras and the fusion properties of these new spin chains.

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Section: The One-boundary Spin Chain and The Blob Algebramentioning

confidence: 90%

$U_q\mathfrak{sl}_2$-invariant non-compact boundary conditions for the XXZ spin chain

Chernyak¹,

Gainutdinov²,

Saleur³

2022

Preprint

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“…After the introduction of the diagrammatic version of the KLR algebra [KhLa09] (see also [Ro08]) they have become quite popular, and might admit handlebody extensions. For example, alongside with KLR algebras Webster's tensor product algebras [Web17], algebras related to Verma categorifications [NV18], [NV17], [MV19], [LNV20], Soergel diagrammatics [EW16], potentially admit handlebody versions, just to name a few.…”

Section: (B)mentioning

confidence: 99%

Handlebody diagram algebras

Tubbenhauer,

Vaz

2021

Preprint

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In this paper we study handlebody versions of classical diagram algebras, most prominently, handlebody versions of Temperley-Lieb, blob, Brauer/BMW, Hecke and Ariki-Koike algebras. Moreover, motivated by Green-Kazhdan-Lusztig's theory of cells, we reformulate the notion of (sandwich, inflated or affine) cellular algebras. We explain this reformulation and how all of the above algebras are part of this theory. Contents1. Introduction 1 2. A generalization of cellularity 4 3. Handlebody braid and Coxeter groups 11 4. Handlebody Temperley-Lieb and blob algebras 16 5. Handlebody Brauer and BMW algebras 28 6. Handlebody Hecke and Ariki-Koike algebras 36 References 41

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“…[ILZ21] or [LV21]. It would be interesting to combine and compare these, also with an eye on categorification of the story as in [LNV21].…”

Section: Introductionmentioning

confidence: 99%

Verma Howe duality and LKB representations

Lacabanne¹,

Tubbenhauer²,

Vaz³

2022

Preprint

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We establish a version of quantum Howe duality with two general linear quantum enveloping algebras that involves a tensor product of Verma modules. We prove that the (colored higher) LKB representations arise from this duality and use this description to show that they are simple as modules for various subgroups of the braid group, including the pure braid group.1A. Schur-Weyl(-Brauer) and Howe dualities. Three main themes in Weyl's seminal book "The classical groups" [Wey97] are the study of polynomial invariants for actions of the eponymous classical groups, and, more or less equivalent, decomposition of the tensor algebra for such an action, and, again more or less equivalent, the description of the invariants in the tensor algebra.The two most prominent examples that fit into Weyl's setting are the celebrated Schur-Weyl duality [Sch01] for tensor invariants of GL m (C) and Brauer duality [Bra37] for tensor invariants of O m (C) and SP m (C) (for the symplectic group m is even). Both of these were studied by using commuting actions of GL m (C), and O m (C), SP m (C) on one side and the symmetric group S n and the Brauer algebra, respectively, on the other side, both acting on a tensor product of the defining representation of the classical groups. In this commuting-action-approach, Schur-Weyl duality for example essentially reads:(A) There are commuting actions of GL m (C) and S n on (C m ) ⊗n .(B) The two actions generate each others centralizer.(C) The GL m (C)-S n bimodule (C m ) ⊗n can be explicitly decomposed into a direct sum of simple GL m (C) modules tensored with simple S n modules. A statement of this form is what we call a double centralizer (a.k.a. double commutant) approach.Howe [How89], [How95] studied polynomial invariants, e.g. via symmetric powers, of classical groups using a double centralizer approach, and the resulting dualities are called Howe dualities

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Tensor product categorifications, Verma modules and the blob 2-category

Cited by 6 publications

References 0 publications

$U_q\mathfrak{sl}_2$-invariant non-compact boundary conditions for the XXZ spin chain

$U_q\mathfrak{sl}_2$-invariant non-compact boundary conditions for the XXZ spin chain

Handlebody diagram algebras

Verma Howe duality and LKB representations

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