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We construct a series of finite-dimensional quantum groups as braided Drinfeld doubles of Nichols algebras of type Super A, for an even root of unity, and classify ribbon structures for these quantum groups. Ribbon structures exist if and only if the rank is even and all simple roots are odd. In this case, the quantum groups have a unique ribbon structure which comes from a non-semisimple spherical structure on the positive Borel Hopf subalgebra. Hence, the categories of finite-dimensional modules over these quantum groups provide examples of non-semisimple modular categories. In the rank-two case, we explicitly describe all simple modules of these quantum groups. We finish by computing link invariants, based on generalized traces, associated to a four-dimensional simple module of the rank-two quantum group. These knot invariants distinguish certain knots indistinguishable by the Jones or HOMFLYPT polynomials.

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We use modified traces to renormalize Lyubashenko’s closed 3-manifold invariants coming from twist non-degenerate finite unimodular ribbon categories. Our construction produces new topological invariants which we upgrade to 2 + 1-TQFTs under the additional assumption of factorizability. The resulting functors provide monoidal extensions of Lyubashenko’s mapping class group representations, as discussed in De Renzi et al. (Commun Contemp Math, 2021. https://doi.org/10.1142/S0219199721500917). This general framework encompasses important examples of non-semisimple modular categories coming from the representation theory of quasi-Hopf algebras, which were left out of previous non-semisimple TQFT constructions.

We construct log-modular quantum groups at even order roots of unity, both as finite-dimensional ribbon quasi-Hopf algebras and as finite ribbon tensor categories, via a de-equivariantization procedure. The existence of such quantum groups had been predicted by certain conformal field theory considerations, but constructions had not appeared until recently. We show that our quantum groups can be identified with those of Creutzig-Gainutdinov-Runkel in type \(A_1\), and Gainutdinov-Lentner-Ohrmann in arbitrary Dynkin type. We discuss conjectural relations with vertex operator algebras at (1, p)-central charge. For example, we explain how one can (conjecturally) employ known linear equivalences between the triplet vertex algebra and quantum \(\mathfrak {sl}_2\), in conjunction with a natural \({{\,\mathrm{PSL}\,}}_2\)-action on quantum \(\mathfrak {sl}_2\) provided by our de-equivariantization construction, in order to deduce linear equivalences between “extended” quantum groups, the singlet vertex operator algebra, and the (1, p)-Virasoro logarithmic minimal model. We assume some restrictions on the order of our root of unity outside of type \(A_1\), which we intend to eliminate in a subsequent paper.

We produce braided commutative algebras in braided monoidal categories by generalizing Davydov’s full center construction of commutative algebras in centers of monoidal categories. Namely, we build braided commutative algebras in relative monoidal centers \( {\mathcal{Z}}_{\mathrm{\mathcal{B}}}\left(\mathcal{C}\right) \) from algebras in ℬ-central monoidal categories \( \mathcal{C} \), where ℬ is an arbitrary braided monoidal category; Davydov’s (and previous works of others) take place in the special case when ℬ is the category of vector spaces \( {\mathbf{Vect}}_{\mathbbm{K}} \) over a field \( \mathbbm{K} \). Since key examples of relative monoidal centers are suitable representation categories of quantized enveloping algebras, we supply braided commutative module algebras over such quantum groups. One application of our work is that we produce Morita invariants for algebras in ℬ-central monoidal categories. Moreover, for a large class of ℬ-central monoidal categories, our braided commutative algebras arise as a braided version of centralizer algebras. This generalizes the fact that centers of algebras in \( {\mathbf{Vect}}_{\mathbbm{K}} \) serve as Morita invariants. Many examples are provided throughout.

We give a new factorisable ribbon quasi-Hopf algebra U, whose underlying algebra is that of the restricted quantum group for sl(2) at a 2p'th root of unity. The representation category of U is conjecturally ribbon-equivalent to that of the triplet vertex operator algebra W(p). We obtain U via a simple current extension from the unrolled restricted quantum group at the same root of unity. The representation category of the unrolled quantum group is conjecturally equivalent to that of the singlet vertex operator algebra M(p), and our construction is parallel to extending M(p) to W(p). We illustrate the procedure in the simpler example of passing from the Hopf algebra for the group algebra CZ to a quasi-Hopf algebra for CZ_{2p}, which corresponds to passing from the Heisenberg vertex operator algebra to a lattice extension.

We show that for dually paired bialgebras, every comodule algebra over one of the paired bialgebras gives a comodule algebra over their Drinfeld double via a crossed product construction. These constructions generalize to working with bialgebra objects in a braided monoidal category of modules over a quasitriangular Hopf algebra. Hence two ways to provide comodule algebras over the braided Drinfeld double (the double bosonization) are provided. Furthermore, a map of second Hopf algebra cohomology spaces is constructed. It takes a pair of 2-cocycles over dually paired Hopf algebras and produces a 2-cocycle over their Drinfeld double. This construction also has an analogue for braided Drinfeld doubles.

This is a survey on Nichols algebras of diagonal type with finite dimension, or more generally with arithmetic root system. The knowledge of these algebras is the cornerstone of the classification program of pointed Hopf algebras with finite dimension, or finite Gelfand-Kirillov dimension; and their structure should be indispensable for the understanding of the representation theory, the computation of the various cohomologies, and many other aspects of finite dimensional pointed Hopf algebras. These Nichols algebras were classified in [H-classif RS] as a notable application of the notions of Weyl groupoid and generalized root system [H-Weyl gpd,HY]. In the first part of this monograph, we give an overview of the theory of Nichols algebras of diagonal type. This includes a discussion of the notion of generalized root system and its appearance in the contexts of Nichols algebras of diagonal type and (modular) Lie superalgebras. In the second and third part, we describe for each Nichols algebra in the list of [H-classif RS] the following basic information: the generalized root system; its label in terms of Lie theory; the defining relations found in [A-jems,A-presentation]; the PBW-basis; the dimension or the Gelfand-Kirillov dimension; the associated Lie algebra as in [AAR2]. Indeed the second part deals with Nichols algebras related to Lie algebras and superalgebras in arbitrary characteristic, while the third contains the information on Nichols algebras related to Lie algebras and superalgebras only in small characteristic, and the few examples yet unidentified in terms of Lie theory.

Let $\mathcal{B}_{\mathfrak{q}}$ be a finite-dimensional Nichols algebra of diagonal type with braiding matrix $\mathfrak{q}$, $\mathcal{L}_{\mathfrak{q}}$ be the corresponding Lusztig algebra as in [ 4], and $\operatorname{Fr}_{\mathfrak{q}}: \mathcal{L}_{\mathfrak{q}} \to U(\mathfrak{n}^{\mathfrak{q}})$ be the corresponding quantum Frobenius map as in [ 5]. We prove that the finite-dimensional Lie algebra $\mathfrak{n}^{\mathfrak{q}}$ is either 0 or the positive part of a semisimple Lie algebra $\mathfrak{g}^{\mathfrak{q}}$, which is determined for each $\mathfrak{q}$ in the list of [ 25].

Given any modular category C over an algebraically closed field k, we extract a sequence (Mg)g≥0 of C-bimodules and show that the Hochschild chain complex CH(C;Mg) of C with coefficients in Mg carries a canonical homotopy coherent projective action of the mapping class group of the surface of genus g+1. The ordinary Hochschild complex of C corresponds to CH(C;M0).
This result is obtained as part of the following more comprehensive topological structure: We construct a symmetric monoidal functor FC:C-Surfc⟶Chk with values in chain complexes over k defined on a symmetric monoidal category of surfaces whose boundary components are labeled with projective objects in C. The functor FC satisfies an excision property which is formulated in terms of homotopy coends. In this sense, any modular category gives naturally rise to a modular functor with values in chain complexes. In zeroth homology, it recovers Lyubashenko's mapping class group representations.
The chain complexes in our construction are explicitly computable by choosing a marking on the surface, i.e. a cut system and a certain embedded graph. For our proof, we replace the connected and simply connected groupoid of cut systems that appears in the Lego-Teichmüller game by a contractible Kan complex.

We show that the category of graded modules over a finite-dimensional algebra admitting a triangular decomposition can be endowed with the structure of a highest weight category. When the algebra is self-injective, we show that this highest weight category has a tilting theory, in the sense of Ringel. As a consequence we are able to show that the degree zero part of the algebra is cellular, whereas, in all cases of interest, the algebra itself is not cellular. The degree zero part is shown to capture all the important information in the original graded category. The heart of the paper is the construction of a highest weight cover of this cellular algebra using a certain finite subquotient of the highest weight category. Thus, beginning with a graded, self-injective algebra admitting a triangular decomposition, we can canonically construct a quasi-hereditary algebra whose representation theory concisely encodes key information, such as the graded multiplicities, of the original algebra. Due to the generality of our approach, the results can be applied to a wide variety of examples, including restricted rational Cherednik algebras, restricted enveloping algebras, Lusztig's small quantum group, hyperalgebras, and finite quantum groups.