Categorification of tensor powers of the vector representation of $U_q(\mathfrak{gl}(1|1))$

Sartori, Antonio

doi:10.48550/arxiv.1305.6162

Cited by 17 publications

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“…Here we recall a categorified action of a variant of gl(1|1) as bimodules over B l (n, k), introduced in [LM19] in the style of [Sar16] and equivalent to a particular case of the bimodules defined in [MR]. We will explain a bordered Floer perspective on the bimodules over B l (n, k), based on Heegaard diagrams, and connect them to deletion and restriction bimodules.…”

Section: Quantum Group Bimodulesmentioning

confidence: 99%

From hypertoric geometry to bordered Floer homology via the m=1 amplituhedron

Lauda,

Licata,

Manion

2020

Preprint

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We give a conjectural algebraic description of the Fukaya category of a complexified hyperplane complement, using the algebras defined in [BLP + 11] from the equivariant cohomology of toric varieties. We prove this conjecture for cyclic arrangements by showing that these algebras are isomorphic to algebras appearing in work of Ozsváth-Szabó [OSz18] in bordered Heegaard Floer homology [LOT18]. The proof of our conjecture in the cyclic case extends work of Karp-Williams [KW19] on sign variation and the combinatorics of the m = 1 amplituhedron. We then use the algebras associated to cyclic arrangements to construct categorical actions of gl(1|1).

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Section: Quantum Group Bimodulesmentioning

confidence: 99%

From hypertoric geometry to bordered Floer homology via the m=1 amplituhedron

Lauda,

Licata,

Manion

2020

Preprint

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“…Canonical bases. We now review the canonical basis of V ⊗n used in [Sart16,LM21] (related to the basis used in [Mani19]). [Sart16].…”

Section: Change-of-basis Bimodulesmentioning

confidence: 99%

“…Basic idempotents of Ozsváth-Szabó's algebras naturally correspond to canonical or crystal basis elements of the underlying representations; in particular, they do not correspond to elements of the standard tensor-product basis. Ellis-Petkova-Vértesi have a related theory with similar properties [PV16,EPV19], called tangle Floer homology; Tian [Tian14] and Sartori [Sart16] have categorifications of the U q (gl(1|1)) representation V ⊗n but do not recover HFK from their constructions. 1 In general, Heegaard Floer homology associates chain complexes to Heegaard diagrams.…”

Section: Introductionmentioning

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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“…Motivated by the strands algebra in bordered Heegaard Floer homology, Khovanov in [18] categorified the positive part of U q (gl(1|2)). A counterpart of our construction in Lie theory is developed by Sartori in [33] using subquotient categories of O(gl n ).…”

Section: Introductionmentioning

confidence: 99%

A categorification of TEXTBACKSLASHmathboldTEXTBACKSLASHUT(𝔰𝔩(1|1)) and its tensor product representations

Yin

2014

Geom. Topol.

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We define the Hopf superalgebra UT (sl(1|1)), which is a variant of the quantum supergroup Uq(sl(1|1)), and its representations V ⊗n 1 for n > 0. We construct families of DG algebras A, B and Rn, and consider the DG categories DGP (A), DGP (B) and DGP (Rn), which are full DG subcategories of the categories of DG A-, Band Rn-modules generated by certain distinguished projective modules. Their 0th homology categories HP (A), HP (B), and HP (Rn) are triangulated and give algebraic formulations of the contact categories of an annulus, a twice punctured disk, and an n times punctured disk. Their Grothendieck groups are isomorphic to UT (sl(1|1)), UT (sl(1|1)) ⊗ Z UT (sl(1|1)) and V ⊗n 1 , respectively. We categorify the multiplication and comultiplication on UT (sl(1|1)) to a bifunctor HP (A) × HP (A) → HP (A) and a functor HP (A) → HP (B), respectively. The UT (sl(1|1))-action on V ⊗n 1 is lifted to a bifunctor HP (A) × HP (Rn) → HP (Rn).1 2 YIN TIANand Rouquier [5] categorified locally finite sl 2 -representations, and more generally, Rouquier [32] constructed a 2-category associated with a Kac-Moody algebra. For the quantum groups themselves, Lauda [22] gave a diagrammatic categorification of U q (sl 2 ) and general cases are given by 20,21]. The program of categorifying Witten-Reshetikhin-Turaev invariants was brought to fruition by Webster [37,38] using the diagrammatic approach.On the other hand, the Alexander polynomial is categorified by knot Floer homology, defined independently by Ozsváth-Szabó [27] and Rasmussen [29]. Although its initial definition was through Lagrangian Floer homology, knot Floer homology admits a completely combinatorial description by . It is natural to ask whether there is a categorical program for U q (sl(1|1)) which is analogous to that of U q (sl 2 ) and which recovers knot Floer homology. This paper presents another step towards such a categorical program. We first define the Hopf superalgebra U T (sl(1|1)) as a variant of U q (sl(1|1)) and the representations V ⊗n 1 of U T (sl(1|1)). Then we categorify the multiplication and comultiplication on U T (sl(1|1)), and its representations V ⊗n 1 . In a subsequent paper [36], we will categorify the action of the braid group on V ⊗n 1 which is induced by the R-matrix structure of U T (sl(1|1)).Our motivation is from the contact category introduced by Honda [8] which presents an algebraic way to study 3-dimensional contact topology. The contact category is closely related to bordered Heegaard Floer homology defined by Lipshitz, Ozsváth and Thurston [23]. See Section 1.3 for more detail on the contact category. Motivated by the strands algebra in bordered Heegaard Floer homology, Khovanov in [18] categorified the positive part of U q (gl(1|2)). A counterpart of our construction in Lie theory is developed by Sartori in [33] using subquotient categories of O(gl n ).

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Categorification of tensor powers of the vector representation of $U_q(\mathfrak{gl}(1|1))$

Cited by 17 publications

References 0 publications

From hypertoric geometry to bordered Floer homology via the m=1 amplituhedron

From hypertoric geometry to bordered Floer homology via the m=1 amplituhedron

Trivalent vertices and bordered knot Floer homology in the standard basis

A categorification of TEXTBACKSLASHmathboldTEXTBACKSLASHUT(𝔰𝔩(1|1)) and its tensor product representations

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