[Paper Review] Quasisymmetric and unipotent tensor categories

Pavel Etingof, Shlomo Gelaki|ArXiv.org|Aug 10, 2007

Algebraic structures and combinatorial models7 references3 citations

TL;DR

This paper classifies quasisymmetric and unipotent tensor categories over ℂ using Drinfeld's quantization method, showing that such categories arise from affine proalgebraic supergroups G with an involution u and a nilpotent element t ∈ (S²𝔤)^G. The key result is a bijection between equivalence classes of quasisymmetric categories and triples (G, u, t), extending Deligne’s theorem to the quasisymmetric setting and classifying coconnected Hopf algebras via quantization of prounipotent Poisson groups.

ABSTRACT

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.

Motivation & Objective

• To extend Deligne’s classification of symmetric tensor categories to quasisymmetric categories, where the square of the braiding is trivial on simple object pairs.
• To classify unipotent tensor categories with fiber functors using quantization of prounipotent Poisson proalgebraic groups.
• To establish a correspondence between coconnected Hopf algebras and data (G, δ) from Lie quasibialgebra structures on prounipotent groups.
• To generalize Drinfeld’s formal deformation theory to categories without a formal parameter ℏ by exploiting termination of power series in nilpotent settings.

Proposed method

• Use Drinfeld’s formal quantization machinery, adapted to nilpotent data, to construct tensor categories from affine proalgebraic supergroups G with an involution u and a nilpotent t ∈ (S²𝔤)^G.
• Apply the universal formula for the product and coproduct in the quantized Hopf algebra O_ℏ(G), which terminates over ℂ[ℏ] due to nilpotency, allowing specialization at ℏ=1.
• Construct the Hopf algebra A(G,δ) over ℂ by specializing the formal deformation O_ℏ(G) at ℏ=1, preserving the coradical filtration and ensuring coconnectedness.
• Use inverse formulas from quantization theory to reconstruct the classical Poisson-Hopf algebra (O(G), μ₀, Δ₀, δ) from a given coconnected Hopf algebra A.
• Leverage Deligne’s theorem on symmetric categories of exponential growth to reduce the classification problem to group-theoretic data.
• Establish a bijection between equivalence classes of quasisymmetric categories and equivalence classes of triples (G, u, t), with t nilpotent and in (S²𝔤)^G.

Experimental results

Research questions

• RQ1 How can Drinfeld’s classification of quasitriangular quasi-Hopf QUE algebras be extended to categories without a formal parameter ℏ?
• RQ2 What is the structure of tensor categories in which the square of the braiding is trivial on all simple object pairs?
• RQ3 Which coconnected Hopf algebras over ℂ arise from quantization of prounipotent Poisson proalgebraic groups?
• RQ4 Can unipotent tensor categories with fiber functors be classified via algebraic data on prounipotent groups?

Key findings

• Equivalence classes of quasisymmetric categories are in bijection with equivalence classes of triples (G, u, t), where G is an affine proalgebraic supergroup, u ∈ G is an involution acting by parity, and t ∈ (S²𝔤)^G is nilpotent.
• The classification of unipotent fiber functors on quasisymmetric categories corresponds bijectively to nilpotent solutions r of the classical Yang-Baxter equation satisfying r + r²¹ = t.
• All coconnected Hopf algebras over ℂ arise as A(G, δ) for some prounipotent group G and Lie quasibialgebra structure δ on 𝔤 = Lie(G), with (G, δ) unique up to isomorphism.
• The quantization of O_ℏ(G) terminates over ℂ[ℏ] due to nilpotency of t, allowing specialization at ℏ=1 to define a Hopf algebra over ℂ.
• The construction preserves the coradical filtration, ensuring that the resulting Hopf algebra A is coconnected, as required.
• The classical limit of a coconnected Hopf algebra recovers a coconnected Poisson-Hopf algebra O(G), which is isomorphic to the original algebra as a vector space.