Abstract
Abstract This paper is a contribution to the construction of non-semisimple modular categories. We establish when Müger centralizers inside non-semisimple modular categories are also modular. As a consequence, we obtain conditions under which relative monoidal centers give (non-semisimple) modular categories, and we also show that examples include representation categories of small quantum groups. We further derive conditions under which representations of more general quantum groups, braided Drinfeld doubles of Nichols algebras of diagonal type, give (non-semisimple) modular categories.
Highlights
The purpose of this article is to establish new constructions of modular tensor categories in the non-semisimple setting
We say that H is quasi-triangular if it comes equipped with an invertible element R “ Rp1q b Rp2q P H b H satisfying p∆ b IdqpRq “ R13R23, pId b ∆qpRq “ R13R12, ∆opphq “ R∆phqR1, for h P H, where ∆op is the opposite coproduct. It follows that H-modpvectkq is a braided tensor category if and only if the finite-dimensional Hopf algebra H is quasi-triangular; here, the braiding is given by cV,W pv b wq “ aW pRp2q b wq b aV pRp1q b vq, for pV, aV q, pW, aW q P H-modpvectkq
We say that H is coquasi-triangular if it comes equipped with a convolution-invertible bilinear form r : HbHÑk satisfying rph, klq “ rphp1q, lqrphp2q, kq, rplh, kq “ rpl, kp1qqrph, kp2qq, rphp1q, lp1qqhp2qlp2q “ lp1qhp1qrphp2q, lp2qq, h, k, l P H. It follows that H-comodpvectkq is a braided tensor category if and only if the finite-dimensional Hopf algebra H is coquasi-triangular; here, the braiding is given by cV,W pv b wq “ pr b IdW b IdV qpIdH b τ b IdV qpδW b δV qpw b vq for pV, δV q, pW, δW q P H-comodpvectkq and τ pa b bq “ b b a
Summary
The purpose of this article is to establish new constructions of modular tensor categories in the non-semisimple setting. Braided Drinfeld double, modular tensor category, Muger centralizer, Nichols algebra relative monoidal center, small quantum group. Consider the relative monoidal center, D :“ ZBpB-modpBqq, or equivalently the category of finite-dimensional modules over the braided Drinfeld double DrinKpB, Bq. D is modular when (i) the canonical symmetric bilinear form b on the coquasi-triangular Hopf algebra K is non-degenerate, and (ii) certain conditions involving elements of the top degree of B and on the dual R-matrix of K are satisfied. We end the paper by constructing, via Proposition 1.7, examples of non-semisimple modular tensor categories attached to Nichols algebras of Cartan type [Example 5.17] and not of Cartan type [Example 5.18] The former includes the representation category of the small quantum group uqpgq at an odd root of unity. The non-semisimple MTCs in Example 5.18 illustrate that our methods can be used to analyze the modularity of representation categories attached to a broader class of Nichols algebras beyond small quantum groups
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