Projective objects and the modified trace in factorisable finite tensor categories

DOI10.1112/S0010437X20007034zbMath1445.18011arXiv1703.00150MaRDI QIDQ5221055

Ingo Runkel, Azat M. Gainutdinov

Publication date: 27 March 2020

Published in: Compositio Mathematica (Search for Journal in Brave)

Full work available at URL: https://arxiv.org/abs/1703.00150

zbMATH Keywords

vertex operator algebrabraided categoryprojective objectpivotal categorymodified traceGro\-then\-dieck ringVerlinde-like formula

Mathematics Subject Classification ID

Vertex operators; vertex operator algebras and related structures (17B69) Projectives and injectives (category-theoretic aspects) (18G05) Grothendieck groups (category-theoretic aspects) (18F30) Monoidal categories, symmetric monoidal categories (18M05) Braided monoidal categories and ribbon categories (18M15)

Related Items (15)

On semisimplicity of module categories for finite non-zero index vertex operator subalgebras ⋮ The symplectic fermion ribbon quasi-Hopf algebra and the \(S L(2, \mathbb{Z})\)-action on its centre ⋮ M-traces in (non-unimodular) pivotal categories ⋮ Mapping class group representations from non-semisimple TQFTs ⋮ Log-modular quantum groups at even roots of unity and the quantum Frobenius I ⋮ Non-semisimple extended topological quantum field theories ⋮ The differential graded Verlinde formula and the Deligne Conjecture ⋮ A quasi-Hopf algebra for the triplet vertex operator algebra ⋮ Frobenius-Perron dimensions of integral \(\mathbb{Z}_+\)-rings and applications ⋮ Modified traces and the Nakayama functor ⋮ Modified traces for quasi-Hopf algebras ⋮ Modified trace is a symmetrised integral ⋮ Further results on the structure of (co)ends in finite tensor categories ⋮ 3-dimensional TQFTs from non-semisimple modular categories ⋮ Extended TQFTs from non-semisimple modular categories

Cites Work

• Higman ideal, stable Hochschild homology and Auslander-Reiten conjecture.
• Generalized trace and modified dimension functions on ribbon categories
• Consistent systems of correlators in non-semisimple conformal field theory
• Symplectic fermions and a quasi-Hopf algebra structure on \(\overline{U}_{\operatorname{i}}s\ell(2)\)
• Holomorphic symplectic fermions
• Indecomposable decomposition of tensor products of modules over the restricted quantum universal enveloping algebra associated to \(\mathfrak{sl}_2\)
• Modular invariance of vertex operator algebras satisfying \(C_2\)-cofiniteness
• Fusion rules and modular transformations in 2D conformal field theory
• Modular group representations and fusion in logarithmic conformal field theories and in the quantum group center
• A \(\mathbb{Z}_{2}\)-orbifold model of the symplectic fermionic vertex operator superalgebra
• Cofiniteness conditions, projective covers and the logarithmic tensor product theory
• Higman's criterion revisited.
• The trace function and Hopf algebras
• Braided groups and quantum Fourier transform
• Representations of finite-dimensional Hopf algebras
• A counterpart of the Verlinde algebra for the small quantum group.
• \(\mathrm{SL}(2, \mathbb{Z})\)-action for ribbon quasi-Hopf algebras
• An equivalence between semisimple symmetric Frobenius algebras and Calabi-Yau categories
• The non-semisimple Verlinde formula and pseudo-trace functions
• On the center of the small quantum group.
• Nonsemisimple fusion algebras and the Verlinde formula
• Modular transformations for tensor categories
• Mapping class group actions on quantum doubles
• Invariants of 3-manifolds and projective representations of mapping class groups via quantum groups at roots of unity
• Traces on ideals in pivotal categories
• The symplectic fermion ribbon quasi-Hopf algebra and the \(S L(2, \mathbb{Z})\)-action on its centre
• Non-degeneracy conditions for braided finite tensor categories
• Invariants of 3-manifolds via link polynomials and quantum groups
• Some remarks on the unrolled quantum group of \(\mathfrak{sl}(2)\)
• Characters and a Verlinde-type formula for symmetric Hopf algebras.
• The monoidal center and the character algebra
• \(\mathbb Z/2\mathbb Z\)-extensions of Hopf algebra module categories by their base categories
• Central ideals and Cartan invariants of symmetric algebras.
• Ambidextrous objects and trace functions for nonsemisimple categories
• The tensor structure on the representation category of the $\mathcal {W}_p$ triplet algebra
• From non-semisimple Hopf algebras to correlation functions for logarithmic CFT
• A BRAIDED MONOIDAL CATEGORY FOR SYMPLECTIC FERMIONS
• VERTEX OPERATOR ALGEBRAS AND THE VERLINDE CONJECTURE
• RIGIDITY AND MODULARITY OF VERTEX TENSOR CATEGORIES
• Modified quantum dimensions and re-normalized link invariants
• Logarithmic conformal field theory, log-modular tensor categories and modular forms
• On Unimodular Finite Tensor Categories
• Modular invariance of characters of vertex operator algebras
• SOME REMARKS ON PSEUDO-TRACE FUNCTIONS FOR ORBIFOLD MODELS ASSOCIATED WITH SYMPLECTIC FERMIONS
• A braided monoidal category for free super-bosons
• Hopf algebras and finite tensor categories in conformal field theory
• From Hopf Algebras to Tensor Categories
• Tensor Categories
• Quantum invariants of knots and 3-manifolds
• Non-semisimple topological quantum field theories for 3-manifolds with corners
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