Remarks on the derived center of small quantum groups

Lachowska, Anna; Qi, You

doi:10.1007/s00029-021-00686-7

Cited by 4 publications

(7 citation statements)

References 32 publications

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“…This gives us the multiplicative structure on Hochschild cohomology. If H is a small quantum group u q (sl 2 ) at a primitive root of unity, the entire multiplicative structure on the Hochschild cohomology is known as graded ring [LQ21]. What Theorem 5.1 tells us in the case of categories of H-modules is how the Gerstenhaber structure on HH * (H) is determined by the canonical structure of H ad as an object in the Drinfeld center of the category of H-modules, i.e.…”

Section: Thenmentioning

confidence: 99%

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Homotopy Invariants of Braided Commutative Algebras and the Deligne Conjecture for Finite Tensor Categories

Schweigert¹,

Woike²

2022

Preprint

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It is easy to find algebras T ∈ C in a finite tensor category C that naturally come with a lift to a braided commutative algebra T ∈ Z(C) in the Drinfeld center of C. In fact, any finite tensor category has at least two such algebras, namely the monoidal unit I and the canonical end X∈C X ⊗ X ∨ . Using the theory of braided operads, we prove that for any such algebra T the homotopy invariants, i.e. the derived morphism space from I to T, naturally come with the structure of a differential graded E 2 -algebra. This way, we obtain a rich source of differential graded E 2 -algebras in the homological algebra of finite tensor categories. We use this result to prove that Deligne's E 2 -structure on the Hochschild cochain complex of a finite tensor category is induced by the canonical end, its multiplication and its non-crossing half braiding. With this new and more explicit description of Deligne's E 2 -structure, we can lift the Farinati-Solotar bracket on the Ext algebra of a finite tensor category to an E 2 -structure at cochain level. Moreover, we prove that, for a unimodular pivotal finite tensor category, the inclusion of the Ext algebra into the Hochschild cochains is a monomorphism of framed E 2 -algebras, thereby refining a result of Menichi.

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Section: Thenmentioning

confidence: 99%

“…Example 6.2 (Quantum groups). Let u q (sl 2 ) be again the small quantum group at a primitive root of unity as discussed in [LQ21]. The Ext algebra of u q (sl 2 ) is computed in [GK93]; it is supported in even degree.…”

Section: Thenmentioning

confidence: 99%

Homotopy Invariants of Braided Commutative Algebras and the Deligne Conjecture for Finite Tensor Categories

Schweigert¹,

Woike²

2022

Preprint

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“…One expects them to correspond to actions of SL(2, Z) on the degree-zero part of the torus state space in TQFT's whose categories of lines are equivalent to modules for various quantum groups (C A-mod with A a quantum group). An SL(2, Z) action on higher Hochschold cohomology of small quantum groups u q (g) at odd roots of unity was constructed in [213], extending [217,218].…”

Section: Hochschild Cohomology Centers and Drinfeld-reshetikhin Mapmentioning

confidence: 99%

“…This part is well studied in the literature; see e.g. [213] and [84,Sec. 7] for similar computations.…”

Section: N Nk and Rectangular W -Algebrasmentioning

confidence: 99%

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A QFT for non-semisimple TQFT

Creutzig¹,

Dimofte²,

Garner³

et al. 2021

Preprint

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We construct a family of 3d quantum field theories T A n,k that conjecturally provide a physical realization -and derived generalization -of non-semisimple mathematical TQFT's based on the modules for the quantum group U q (sl n ) at an even root of unity q = exp(iπ/k). The theories T A n,k are defined as topological twists of certain 3d N = 4 Chern-Simons-matter theories, which also admit string/M-theory realizations. They may be thought of as SU (n) k−n Chern-Simons theories, coupled to a twisted N = 4 matter sector (the source of non-semisimplicity). We show that T A n,k admits holomorphic boundary conditions supporting two different logarithmic vertex operator algebras, one of which is an sl n -type Feigin-Tipunin algebra; and we conjecture that these two vertex operator algebras are related by a novel logarithmic level-rank duality. (We perform detailed computations to support the conjecture.) We thus relate the category of line operators in T A n,k to the derived category of modules for a boundary Feigin-Tipunin algebra, and -using a logarithmic Kazhdan-Lusztiglike correspondence that has been established for n = 2 and expected for general n -to the derived category of U q (sl n ) modules. We analyze many other key features of T A n,k and match them from quantum-group and VOA perspectives, including deformations by flat P GL(n, C) connections, one-form symmetries, and indices of (derived) genus-g state spaces.

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Homotopy invariants of braided commutative algebras and the Deligne conjecture for finite tensor categories

Schweigert

Woike

2023

Advances in Mathematics

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Remarks on the derived center of small quantum groups

Cited by 4 publications

References 32 publications

Homotopy Invariants of Braided Commutative Algebras and the Deligne Conjecture for Finite Tensor Categories

Homotopy Invariants of Braided Commutative Algebras and the Deligne Conjecture for Finite Tensor Categories

A QFT for non-semisimple TQFT

Homotopy invariants of braided commutative algebras and the Deligne conjecture for finite tensor categories

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