[Paper Review] On the Central Charge of a Factorizable Hopf Algebra

Yorck Sommerhaeuser, Yongchang Zhu|arXiv (Cornell University)|Jun 18, 2009

Algebraic structures and combinatorial models29 references3 citations

TL;DR

This paper establishes that for a semisimple factorizable Hopf algebra over a field of characteristic zero, the central charge is always an integer, with the value of the integral on the Drinfel'd element and its inverse differing by at most a fourth root of unity. When the dimension is odd, the central charge is an even integer, determined precisely by the dimension modulo 4.

ABSTRACT

For a semisimple factorizable Hopf algebra over a field of characteristic zero, we show that the value that an integral takes on the inverse Drinfel'd element differs from the value that it takes on the Drinfel'd element itself at most by a fourth root of unity. This can be reformulated by saying that the central charge of the Hopf algebra is an integer. If the dimension of the Hopf algebra is odd, we show that these two values differ at most by a sign, which can be reformulated by saying that the central charge is even. We give a precise condition on the dimension that determines whether the plus sign or the minus sign occurs. To formulate our results, we use the language of modular data.

Motivation & Objective

• To determine the precise nature of the central charge in semisimple factorizable Hopf algebras over characteristic zero fields.
• To investigate whether the central charge is an even integer when the dimension of the Hopf algebra is odd.
• To establish a general condition on the dimension that determines the sign of the ratio between the integral values on the Drinfel'd element and its inverse.
• To generalize classical results on Gaussian sums to the setting of modular data and Hopf algebras.
• To provide a framework using modular data that isolates the essential algebraic properties needed for the central charge results.

Proposed method

• Uses modular data consisting of Verlinde matrix S and Dehn matrix T, with rescaling freedom to simplify relations.
• Applies Galois group actions and cyclotomic field theory to analyze symmetries of the modular data.
• Employs group cohomology and Beyl’s theorem to study relations in the modular group action.
• Uses Cauchy’s theorem and properties of roots of unity to analyze the order of central charge-related quantities.
• Reduces the problem to verifying that certain matrix images satisfy the defining relations of SL(2, Z/nZ) for n = 3, 8, or 24.
• Leverages the fact that the ratio g/g′ is a root of unity, and uses normalization to express this as an exponential of the central charge c.

Experimental results

Research questions

• RQ1 Does the central charge of a semisimple factorizable Hopf algebra over a field of characteristic zero always lie in the integers?
• RQ2 If the dimension of the Hopf algebra is odd, is the central charge necessarily an even integer?
• RQ3 What determines whether the integral on the Drinfel'd element and its inverse differ by +1 or −1 in the odd-dimensional case?
• RQ4 Can the classical result on Gaussian sums, where G² = ±n, be generalized to the setting of modular categories and Hopf algebras?
• RQ5 What is the precise role of the fourth root of unity in the general case, and how does it constrain the central charge?

Key findings

• For any semisimple factorizable Hopf algebra over a field of characteristic zero, the central charge is an integer, as g⁴ = g′⁴ holds.
• When the dimension n is odd, the central charge is even: g′ = g if n ≡ 1 mod 4, and g′ = −g if n ≡ 3 mod 4.
• The sign of the ratio between the integral on the Drinfel'd element and its inverse is determined solely by the dimension modulo 4.
• The result generalizes classical results on Gaussian sums, where G² = ±n, to the setting of Hopf algebras and modular categories.
• The central charge c satisfies e^{2πic/24} = ℓ, and the kernel of the associated representation contains Γ(8) or Γ(24), depending on normalization.
• The framework using modular data allows the results to be derived independently of the Hopf algebra structure, highlighting the role of intrinsic algebraic symmetries.