## Hopf Algebras and Congruence Subgroups

### Yorck Sommerhäuser
Yongchang Zhu

- Preprint: XXX preprint archive: math.RA/0710.0705
- Book: Mem. Am. Math. Soc., Vol. 219, No. 1028, Am. Math. Soc., Providence, 2012

#### Abstract

We prove that the kernel of the natural action of the modular group on the center of the Drinfel'd double of a semisimple Hopf algebra is a congruence subgroup. To do this, we introduce a class of generalized Frobenius-Schur indicators and endow it with an action of the modular group that is compatible with the original one.

#### Introduction

At least since the work of J. L. Cardy in 1986, the importance of the role of the modular group has been emphasized in conformal field theory, and it has been extensively investigated since then.^{1} This importance stems from the fact that the characters of the primary fields, which depend on a complex parameter, are equivariant with respect to the action of the modular group on the upper half plane on the one hand and a linear representation of the modular group on the other hand, which is finite-dimensional in the case of a rational conformal field theory. It was soon noticed in the course of this development that under quite general assumptions a frequently used generator of the modular group has finite order in this representation.^{2} Since this generator and one of its conjugates together generate the modular group, this leads naturally to the conjecture that the kernel of the before-mentioned representation is a congruence subgroup. After an intense investigation, this conjecture was finally established by P. Bantay.^{3}

In a different line of thought, Y. Kashina observed, while investigating whether the antipode of a finite-dimensional Yetter-Drinfel'd Hopf algebra over a semisimple Hopf algebra has finite order, that certain generalized powers associated with the semisimple Hopf algebra tend to become trivial after a certain number of steps.^{4} She established this fact in several cases and conjectured that in general this finite number after which the generalized powers become trivial, which is now called the exponent of the Hopf algebra, divides the dimension of the Hopf algebra. This conjecture is presently still open. However, P. Etingof and S. Gelaki, realizing the connection between these two lines of thought, were able to establish the finiteness of the exponent and showed that it divides at least the third power of the dimension.^{5} They also explained the connection of the exponent to the order of the generator of the modular group by showing that the exponent of the Hopf algebra is equal to the order of the Drinfel'd element of the Drinfel'd double of the Hopf algebra. In this context, it should be noted that this connection between Hopf algebras and conformal field theory has been intensively investigated by many authors; we only mention here the modular Hopf algebras and modular categories of N. Reshetikhin and V. G. Turaev on the one hand and the modular transformations considered by V. Lyubashenko and his coauthors on the other hand.^{6}

It is the purpose of the present work to unite these two lines of thought further by establishing an analogue of Bantay's results for semisimple Hopf algebras. We will show in Theorem 8.3 that the kernel of the action of the modular group on the center of the Drinfel'd double of a semisimple Hopf algebra is a congruence subgroup of level N, where N is the exponent of the Hopf algebra discussed above. The proof of this theorem becomes possible by the use of a new tool, a further generalization of the higher Frobenius-Schur indicators studied earlier by Y. Kashina and the authors.^{7} These new indicators, which we call equivariant Frobenius-Schur indicators, are functions on the center of the Drinfel'd double and carry an action of the modular group that is equivariant with respect to the action of the modular group on the center. This equivariance in particular connects, via the action of the Verlinde matrix that arises from the other frequently used generator of the modular group, the first formula for the higher Frobenius-Schur indicators with the second resp. third formula, whose interplay is crucial for the proof of Cauchy's theorem for Hopf algebras.^{8}

The Drinfel'd double is an example of a factorizable Hopf algebra, and the results for the Drinfel'd double can be partially generalized to this more general class. However, in the case of a factorizable semisimple Hopf algebra, the modular group acts in general only projectively on the center of the Hopf algebra. This phenomenon also occurs in conformal field theory, and also in the general framework of modular categories, of which the representation category of a semisimple factorizable Hopf algebra is an example.^{9}
But it is still possible to talk about the kernel of the projective representation, i.e., the subgroup of the modular group that acts as the identity on the associated projective space of the center. We will also show, in Paragraph 8.4, that in this more general case this so-called projective kernel is a congruence subgroup of level N.

The article is organized as follows: In Section 1, after briefly recalling some facts about the modular group, we describe a relation that characterizes the orbits of the principal congruence subgroups and plays an important role in the proof of the orbit theorem in Paragraph 7.6. In Section 2, we recall some basic facts about quasitriangular Hopf algebras and the Drinfel'd double construction, and prove some lemmas about the Drinfel'd element and the evaluation form. In Section 3, we prove some facts about factorizable Hopf algebras that are important for the equivariance properties that we will discuss later. In Section 4, we construct the action of the modular group on the center of a factorizable Hopf algebra. It must be emphasized that this construction is not new; on the contrary, it is discussed in abundance in the literature we have already quoted, especially in V. G. Turaev's monograph on the one hand and in two closely related articles V. Lyubashenko on the other hand.^{10} What we do in this section is to translate Lyubashenko's graphical proof of the modular identities into the language of quasitriangular Hopf algebras, thereby offering a presentation of these results that is not yet available in the literature in this form.^{11}

In Section 5, we specialize to the semisimple case. We can then use the centrally primitive idempotents as a basis and therefore get explicit matrices for the action of the modular group constructed in Section 4. In the case of a Drinfel'd double, there is a different construction for the action of the modular group based on the evaluation form and using a slightly less frequently used set of generators of the modular group. This description of the action, which is crucial for the proof of the equivariance theorem in Paragraph 7.5, is given in Section 6.
In Section 7, we introduce the equivariant Frobenius-Schur indicators I_{V}((m,l),z), which depend on an H-module V, two integers m and l, and a central element z in the Drinfel'd double D(H). We prove the equivariance theorem
I_{V}((m,l)g,z) = I_{V}((m,l),g.z) for an element g of the modular group.
In Paragraph 7.6, we prove the orbit theorem, which asserts that the equivariant indicators only depend on the orbit of (m,l) under the principal congruence subgroup determined by the exponent. This is applied in Section 8 to prove the congruence subgroup theorem, which asserts that g.z=z for all z in the center of the Drinfel'd double D(H) and all g in the principal congruence subgroup. Note that the orbit theorem is an immediate consequence of the equivariance theorem and the congruence subgroup theorem. Finally, in the case of an arbitrary factorizable Hopf algebra, we prove the projective congruence subgroup theorem, which asserts that the kernel of the projective representation is a congruence subgroup.

Throughout the whole exposition, we consider an algebraically closed base field that is denoted by K. From Section 5 on until the end, we assume in addition that K has characteristic zero. All vector spaces considered are defined over K, and all tensor products without subscripts are taken over K. The dual of a vector space V is denoted by V^{*}:=Hom_{K}(V,K), and the transpose of a linear
map f: V → W is denoted by f^{*}: W^{*} → V^{*}.
Unless stated otherwise, a module is a left module. Also, we use the so-called Kronecker symbol δ_{ij}, which is equal to 1 if i=j and zero otherwise. The set of natural numbers is the set ℕ
:={1,2,3,...}; in particular, 0 is not a natural number. The symbol ℚ_{m} denotes the m-th cyclotomic field, and not the field of m-adic numbers, and ℤ_{m} denote the set ℤ/mℤ of integers modulo m, and not the ring of m-adic integers.
The greatest common divisor of two integers m and l is denoted by gcd(m,l) and is always chosen to be nonnegative.

Furthermore, H denotes a Hopf algebra of finite dimension n with coproduct Δ, counit ε, and antipode S. We will use the same symbols to denote the corresponding structure elements of the dual Hopf algebra H^{*}, except for the antipode, which is denoted by S^{*}. The opposite Hopf algebra, in which the multiplication is reversed, is denoted by H^{op}, and the coopposite Hopf algebra, in which the comultiplication is reversed, is denoted by H^{cop}. If b_{1},...,b_{n} is a basis of H with dual basis b_{1}^{*},...,b_{n}^{*}, we have the formulas^{11}

_{i=1}

^{n}b

_{i}

^{*}⊗ b

_{i(1)}⊗ b

_{i(2)}⊗ ... ⊗ b

_{i(m)}= ∑

^{n}

_{i1,i2,...,im=1}b

_{i1}

^{*}b

_{i2}

^{*}... b

_{im}

^{*}⊗ b

_{i1}⊗ b

_{i2}⊗ ... ⊗ b

_{im}

and

_{i=1}

^{n}b

_{i(1)}

^{*}⊗ b

_{i(2)}

^{*}⊗ ... ⊗ b

_{i(m)}

^{*}⊗ b

_{i}= ∑

^{n}

_{i1,i2,...,im=1}b

_{i1}

^{*}⊗ b

_{i2}

^{*}⊗ ... ⊗ b

_{im}

^{*}⊗ b

_{i1}b

_{i2}... b

_{im}

which we will refer to as the dual basis formulas. We use the letter A instead of H if the Hopf algebra under consideration is quasitriangular. With respect to enumeration, we use the convention that propositions, definitions, and similar items are referenced by the paragraph in which they occur; they are only numbered separately if this reference is ambiguous.

The essential part of the present work was carried out when the first author held a visiting research position at the Hong Kong University of Science and Technology. He thanks the university, and in particular his host, for the hospitality. He also thanks the University of Cincinnati for a follow-on visiting position during which part of the manuscript was written. The second author would like to express his appreciation for the support by the RGC Competitive Earmarked Research Grant HKUST 6059/04.

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