Some remarks on the unrolled quantum group of sl(2)

Abstract

In this paper we consider the representation theory of a non-standard quantization of sl(2). This paper contains several results which have applications in quantum topology, including the classification of projective indecomposable modules and a description of morphisms between them. In the process of proving these results the paper acts as a survey of the known representation theory associated to this non-standard quantization of sl(2). The results of this paper are used extensively in [arXiv:1404.7289] to study Topological Quantum Field Theory (TQFT) and have connections with Conformal Field Theory (CFT).

25 pages, v3: several mistakes corrected in the formulas of modified traces

References in corpus (5)

• Coset Constructions of Logarithmic (1,p)-Models
• Logarithmic intertwining operators and W(2,2p-1)-algebras
• Lusztig limit of quantum sl(2) at root of unity and fusion of (1,p) Virasoro logarithmic minimal models
• Trace decategorification of categorified quantum sl(2)
• Relations between Witten-reshetikhin-turaev and non semi-simple sl(2) 3-manifold invariants

Cited by in corpus (33)

• Logarithmic conformal field theory, log-modular tensor categories and modular forms
• A quasi-Hopf algebra for the triplet vertex operator algebra
• On ribbon categories for singlet vertex algebras
• The trace on projective representations of quantum groups
• Braided Tensor Categories related to ${\mathcal{B}}_p$ Vertex Algebras
• Logarithmic Hennings invariants for restricted quantum sl(2)
• Direct limit completions of vertex tensor categories
• Non-Semisimple Extended Topological Quantum Field Theories
• Relations between Witten-reshetikhin-turaev and non semi-simple sl(2) 3-manifold invariants
• Tensor structure on the Kazhdan-Lusztig category for affine $\mathfrak{gl}(1|1)$
• Asymptotic behavior of partial and false theta functions arising from Jacobi forms and regularized characters
• Ribbon tensor structure on the full representation categories of the singlet vertex algebras
• Projective objects and the modified trace in factorisable finite tensor categories
• Logarithmic conformal field theories of type $B_n,\ell =4$ and symplectic fermions
• Quantum SL(2) and logarithmic vertex operator algebras at (p,1)-central charge
• Logarithmic Link Invariants of ${\overline{U}}_q^H({\mathfrak{sl}}_2)$ and Asymptotic Dimensions of Singlet Vertex Algebras
• Characterizing braided tensor categories associated to logarithmic vertex operator algebras
• Modular Categories and TQFTs Beyond Semisimplicity
• Modified traces for quasi-Hopf algebras
• Modified graded Hennings invariants from unrolled quantum groups and modified integral
• On Regularised Quantum Dimensions of the Singlet Vertex Operator Algebra and False Theta Functions
• $U_q{\mathfrak{sl}}_2$-invariant non-compact boundary conditions for the XXZ spin chain
• Pseudo-Hermitian Levin-Wen models from non-semisimple TQFTs
• A Hermitian TQFT from a non-semisimple category of quantum sl(2)-modules
• Universal quantum computation using Ising anyons from a non-semisimple Topological Quantum Field Theory
• Quantum holonomic link invariants derived from stated skein algebras
• Categories of Weight Modules for Unrolled Restricted Quantum Groups at Roots of Unity
• A Hennings type invariant of $3$-manifolds from a topological Hopf superalgebra
• Modified trace from pivotal Hopf $G$-coalgebra
• Quantum Invariants of 3-Manifolds Arising from Non-Semisimple Categories
• On unrolled Hopf algebras
• Verma Modules for Restricted Quantum Groups at a Fourth Root of Unity
• Stated skein algebras and their representations