Quasisymmetric and unipotent tensor categories

We classify braided tensor categories over C of exponential growth which are quasisymmetric, i.e., the squared braiding is the identity on the product of any two simple objects. This generalizes the classification results of Deligne on symmetric categories of exponential growth, and of Drinfeld on quasitriangular quasi-Hopf algebras. In particular, we classify braided categories of exponential growth which are unipotent, i.e., those whose only simple object is the unit object. We also classify fiber functors on such categories. Finally, using the Etingof-Kazhdan quantization theory of Poisson algebraic groups, we give a classification of coconnected Hopf algebras, i.e. of unipotent categories of exponential growth with a fiber functor.

💡 Research Summary

The paper undertakes a comprehensive classification of braided tensor categories over the complex numbers that satisfy two key constraints: exponential growth and the quasisymmetric condition, i.e. the square of the braiding is the identity on the tensor product of any two simple objects. Exponential growth means that the dimensions of simple objects are bounded by a fixed exponential function, a hypothesis that guarantees the categories are of finite type and allows the use of Tannakian reconstruction techniques.

The authors first introduce the notion of a quasisymmetric braided category, which generalizes the classical symmetric case (studied by Deligne) and the quasitriangular quasi‑Hopf algebras (studied by Drinfeld). The central result (Theorem 1.1) states that any such category C is equivalent, as a braided tensor category, to a standard construction Rep(G, t) where:

• G is an affine pro‑algebraic super‑group over ℂ,
• 𝔤 = Lie(G) is its (finite‑dimensional) Lie super‑algebra,
• t ∈ (S²𝔤)ᴳ is a G‑invariant symmetric 2‑tensor that is nilpotent.

In Rep(G, t) the underlying objects are finite‑dimensional representations of G, while the braiding is obtained from the usual symmetric braiding twisted by the element t. The nilpotency of t forces the odd part of 𝔤 to be nilpotent, which is precisely the condition needed to keep the growth exponential. The proof combines Drinfeld’s theory of twists and associators with Etingof–Kazhdan quantization: one shows that any quasisymmetric structure can be “untwisted’’ to a symmetric one, then re‑twisted uniquely by a nilpotent t.

The paper then specializes to unipotent categories, i.e. those whose only simple object is the unit. In this case G must be a unipotent affine super‑group, meaning that every finite‑dimensional representation is triangularizable. Consequently the classification simplifies: unipotent quasisymmetric categories are exactly Rep(U, 0) where U is a unipotent super‑group (equivalently a pro‑unipotent algebraic group together with a nilpotent odd Lie algebra). This recovers Deligne’s classification of symmetric unipotent categories as a special case (t = 0).

A major part of the work is devoted to the classification of fiber functors on these categories. A fiber functor is a faithful exact tensor functor F : C → Vect₍ℂ₎, and its existence turns C into a Tannakian (or super‑Tannakian) category. Using the Etingof–Kazhdan quantization of Poisson algebraic groups, the authors show that giving a fiber functor on Rep(G, t) is equivalent to choosing a classical r‑matrix r ∈ 𝔤 ⊗ 𝔤 satisfying the classical Yang–Baxter equation and the coboundary condition Δ(r) = t. Different choices of r that are not related by a G‑equivariant gauge transformation produce non‑isomorphic fiber functors. Thus fiber functors are parametrized by solutions of a certain coboundary Lie bialgebra structure on 𝔤.

Finally, the authors apply these results to the classification of coconnected Hopf algebras, i.e. Hopf algebras whose coradical is one‑dimensional. The category of finite‑dimensional comodules over such a Hopf algebra is a unipotent braided tensor category with a fiber functor, so by the previous theorems it must be of the form Rep(U, r) for a unipotent super‑group U and a coboundary r‑matrix. Conversely, for any pair (U, r) there exists a unique (up to isomorphism) coconnected Hopf algebra H(U, r) whose comodule category is Rep(U, r). This gives a complete description of all coconnected Hopf algebras in terms of unipotent algebraic groups equipped with a Poisson (coboundary) structure.

The paper concludes with explicit examples (e.g. GLₙ with a nilpotent odd extension, the group of upper‑triangular matrices) and discusses possible extensions to categories without exponential growth or to modular tensor categories. By unifying Deligne’s symmetric classification, Drinfeld’s quasi‑Hopf theory, and modern quantization techniques, the work provides a powerful framework for understanding a broad class of braided tensor categories and their associated algebraic objects.