- Preprint: XXX Preprint-Archiv: math.RA/0906.3471
- Zeitschrift: Adv. Math. 236 (2013), 158–223

Every semisimple factorizable Hopf algebra, or more generally every modular category, gives rise to numbers g and g' that are in many ways analogous to the Gaussian sum G and the reciprocal sum G'. In fact, as we will discuss in Paragraph 4.4, both sets of numbers become equal for a suitable choice of the factorizable Hopf algebra, so that the results for Hopf algebras can be viewed as direct generalizations of the results for Gaussian sums. Therefore, looking at the above equations, the question arises whether we also have the equations gg' = n and g' = ± g. Proofs of the first of these two equations can be found in many places in the literature; we mention here only [3], Cor. 3.1.11, p. 52 and [73], Exerc. II.5.6, p. 116. For the second equation, less is known: A result known as Vafa's theorem implies that g/g' is a root of unity (cf. [3], Thm. 3.1.19, p. 57). It is customary to write this quotient as an exponential of a parameter c called the central charge, using certain normalization conventions that we will explain in Paragraph 1.7. Working through these conventions, the fact that g/g' is a root of unity translates into the condition that the central charge is a rational number, this fact being at least one of the reasons for the appearance of the word 'rational' in the term 'rational conformal field theory'.

However, the question that arose from the classical case above was whether we have g' = ± g, which translates into the condition that the central charge is an even integer. The first main result of this article is
that this is indeed the case if the dimension of the Hopf algebra is odd.
We conjecture that the result still holds if the dimension of the Hopf algebra is even. But what we can prove, and this is the second main result of this article, is that we have g^{4} = g'^{4}, which means for the central charge that it is an integer.

Let us be more specific. From the R-matrix of a semisimple factorizable
Hopf algebra over a field of characteristic zero, we can derive a special element, the so-called Drinfel'd element u. The analogue of the Gaussian sum G that we are considering is the
number g = χ_{R}(u^{-1}), where χ_{R} denotes the character of the regular representation. Note that in this situation χ_{R} can also be described as the unique two-sided integral that satisfies χ_{R}(1)=n,
where n denotes the dimension of the Hopf algebra. Similarly, the reciprocal Gaussian sum has the form g'=χ_{R}(u). In the case where n is odd, our first main result is that
g' = g if n ≡ 1 (mod 4), and g' = -g if n ≡ 3 (mod 4), so that g^{2}=g'^{2}.
We conjecture that this is still correct if n is even, but we can only
prove that g^{4}=g'^{4} in this case, which is our second main result, both of which together form exactly the two parts of
Theorem 4.3.

All these results can be stated and proved entirely within the Hopf algebra framework. However, we have chosen not to do this, but rather to take an axiomatic approach via modular data. This has at least three advantages: It is considerably more elementary, it makes clearer which properties are actually used to establish the results, and it facilitates the transfer of the results from the theory of Hopf algebras to related fields.

A modular datum essentially consists of two matrices that satisfy certain relations. Although these matrices and the relations between them pervade the entire literature on conformal field theory, it appears that the first systematic attempt to cast these properties into a set of simple axioms has been undertaken by T. Gannon (cf. [26], [27]). Our definition in Paragraph 1.1 is a minor variant of his. In particular, we have designed our definition in such a way that the two essential matrices can be rescaled rather arbitrarily, and we have given names to the matrices: We speak of the Verlinde matrix S and of the Dehn matrix T. We have done this because the symbols used for them vary quite a bit over the literature, so that it appears inappropriate to speak of the S-matrix and the T-matrix, in particular as this usage is exactly opposite to the original conventions, as used for example by R. Fricke and F. Klein (cf. [40], § II.2.2, p. 209).

Instead of using natural numbers 1,2,3,... to index the rows and the columns of our matrices, we have used an abstract finite index set I. The reason for this is that the basic operation for modular data is to take their Kronecker product, and the corresponding new index set, the Cartesian product of the two old ones, is not canonically modeled on a string of natural numbers. This operation is important because it forms the basis of a generalized Witt group for modular data, and, as in the classical case, the Gaussian sum is a homomorphism with respect to this group structure. However, we do not elaborate on this aspect in the sequel, at least not in this article.

Properties of modular data that break the scalability are given special names: We call a modular datum normalized if the elements s_{oo} and t_{o} in the left upper corner of the Verlinde and the Dehn matrix are equal to 1. If the elements in the first column of the Verlinde matrix are positive integers, we call the modular datum integral. It is mainly this integrality property that distinguishes modular data coming from semisimple factorizable Hopf algebras from other modular data.

It is built into the axioms that every modular datum gives rise to a projective representation of the homogeneous modular group. By definition, rescaling the Verlinde and the Dehn matrix does not change this projective representation. Now every projective representation of the modular group can be lifted to an ordinary linear representation; as we discuss in Paragraph 1.7, this is a direct consequence of the defining relations. To do this, however, we need to fix the scaling of the Verlinde and the Dehn matrix, which means that we have to choose two parameters, called the generalized rank D and the multiplicative central charge l, a close relative of the additive central charge c considered above. A modular datum for which these two choices have been made is called an extended modular datum. As we illustrate in Paragraph 5.6 by an instructive example, the choice of these two parameters greatly affects the properties of the extended modular datum. After the two parameters have been chosen, they can be used to scale the Verlinde and the Dehn matrix correctly, so that we get an ordinary representation of the modular group; these correctly scaled matrices are called the homogeneous Verlinde matrix S' and the homogeneous Dehn matrix T'. In this context, let us note that it is customary in the literature to require that the square of the parameter D is equal to n, the global dimension of the modular datum; however, the only property that is necessary for the lifting is the condition D^{4}=n^{2}. For this reason,
we distinguish in Paragraph 1.7 between a rank and a generalized rank.

These general aspects of modular data that we have just discussed form the
contents of the preliminary Section 1. We do not claim any
originality here; we have just rearranged known material in a way that suits our later needs. The first section that contains new results is Section 2. We begin by introducing three notions:
We say what a congruence datum and a projective congruence datum is,
and define a Galois datum to be an integral modular datum whose entries t_{i} of the Dehn matrix are compatible with a certain action of a
Galois group described in Paragraph 1.4 via the compatibility
condition t_{σ.i} = σ^{2}(t_{i}). We then concentrate on the case where the exponent N is odd, and reach after several auxiliary steps our first main result, namely Theorem 2.5: If the exponent of a normalized integral modular datum is odd, we have g' = ± g. If the global dimension n of the modular datum is also odd, then we can sharpen this assertion and obtain, by comparing with the classical Gaussian sum G, the result that g' = g if n ≡ 1 (mod 4)
and g' = - g if n ≡ 3 (mod 4), which is stated in Theorem 2.6. Recall that for Hopf algebras, which are still
our main concern, it is a special case of Cauchy's theorem (cf. [37],
Thm. 3.4, p. 26) that n is odd if and only if N is odd.
If we extend the modular datum using a rank, then Theorem 2.6 means for the corresponding additive central charge c
that it is an even integer satisfying c ≡ 0 (mod 4) if n ≡ 1 (mod 4) and c ≡ 2 (mod 4) if n ≡ 3 (mod 4).

In Section 3, we turn to the case where N is even. Our main tool here is a result by F. R. Beyl, which we quote in Paragraph 3.1 and which asserts that the Schur multiplicator
of the reduced modular group SL(2,Z_{N}) is either trivial or isomorphic to Z_{2}. As we explain, it follows from this that, in contrast to the case of the unreduced modular group considered above, projective representations of the reduced modular group cannot always be completely lifted to an ordinary representation, but can be lifted up to a sign. In fact, we will see a nice explicit example of this phenomenon in Paragraph 5.6. From Beyl's result and a couple of auxiliary steps,
we deduce in Theorem 3.4 that, for a normalized extended projective congruence datum that is also Galois, we have g^{4}=g'^{4}. For the multiplicative central charge, this means that l^{24}=1, which in turn means that the additive central charge c is an integer. As the example just mentioned also shows, the conclusion of this theorem cannot be sharpened to the assertion that g^{2}=g'^{2}, as in the case of odd exponents. However, in the case of Hopf algebras, we conjecture that it can be sharpened in this way.

It is very important to emphasize that the same result, the integrality of c, has been already derived before by A. Coste and T. Gannon from assumptions that look, and are, very similar (cf. [11], § 2.4, Prop. 3.b, p. 9). Their argument is short and elegant, and we also use it as a part of our proof, as the reader will confirm when looking at Lemma 2.3. However, in the preceding terminology, Coste and Gannon impose the Galois condition on the homogeneous Dehn matrix T', whereas we, having the applications to Hopf algebras in mind, impose it on the original Dehn matrix T. To deduce their assumption from ours is therefore exactly equivalent to proving our result, as we explain in greater detail in Paragraph 3.4.

Section 3 ends with some results on the relation of the prime divisors of N and the prime divisors of n. This is, of course, inspired by Cauchy's theorem for Hopf algebras mentioned above. In Corollary 3.5, we prove in particular that, for a projective congruence datum that is also Galois, we have N ≡ 0 (mod 4) if n ≡ 2 (mod 4). We do not think, however, that this result is the best possible one, and close the section with a conjecture about a potential improvement.

Finally, we apply in Section 4 all the machinery developed before to Hopf algebras. What we consider are semisimple factorizable Hopf algebras over fields of characteristic zero, which lead, as we explain, via the Drinfel'd element u to modular categories and integral modular data. The application of the preceding results is now rather straightforward. If the dimension n of the Hopf algebra is odd, Cauchy's theorem and Theorem 2.6 now yield that
χ_{R}(u) = χ_{R}(u^{-1}) if n ≡ 1 (mod 4) and χ_{R}(u) = - χ_{R}(u^{-1}) if n ≡ 3 (mod 4). On the other hand, if n is even, Theorem 3.4 yields that χ_{R}(u)^{4} = χ_{R}(u^{-1})^{4}. Of course, we have to verify the rather strong assumptions of this theorem, but here we can rely
on earlier work: That we are working with a projective congruence datum
follows from the projective congruence subgroup theorem for Hopf algebras
(cf. [68], Thm. 9.4, p. 94), and it is also known that our modular datum is Galois (cf. [68], Lem. 12.2, p. 115). All these
results are obtained in Paragraph 4.3, where we also state
explicitly our conjecture that the
equation χ_{R}(u)^{2} = χ_{R}(u^{-1})^{2} also holds in the case of an even-dimensional Hopf algebra. We also expect that the dimension of such an even-dimensional semisimple factorizable Hopf algebra is always divisible by 4.

At the end of Section 4, we discuss an example that was constructed originally by D. E. Radford (cf. [60], Sec. 3, p. 10; [61], Sec. 2.1, p. 219). In this example, the group ring of a cyclic group of odd order n is endowed with a nonstandard R-matrix based on a primitive n-th root of unity. For this R-matrix, the Gaussian sum g becomes the classical Gaussian sum G, which shows that our preceding considerations really are generalizations of the classical case.

In the same way as Hopf algebras, quasi-Hopf algebras lead to modular data,
and these are also integral if their ribbon element is chosen correctly.
Therefore, our methods apply in just the same way, as we discuss
in Section 5. However, our results in the case of a quasi-Hopf algebra are less strong, which is due to the fact that, although the projective congruence subgroup theorem has been carried over to quasi-Hopf algebras (cf. [57], Thm. 8.8, p. 35), the Galois property of the corresponding modular datum has not yet been established. We are, however, optimistic that this will happen in the near future, and then our methods can be used to deduce that we have g^{4}=g'^{4} also in the quasi-Hopf algebra case. But this result can, and this is the main point of the discussion, not be improved: We construct
an explicit example of a quasi-Hopf algebra for which
g^{2} ≠ g'^{2}.
As we mentioned above, we conjecture that it is impossible to construct
such an example with an ordinary Hopf algebra.

Both this quasi-Hopf algebra and the arising modular datum have been
considered before several times; however, we think that at least explicitly the connection between the two has not been made so far. Using a term from [63], we call the modular datum the semion datum; it is usually constructed using affine Kac-Moody algebras, and there
is a particularly simple case. The quasi-Hopf algebra that we are using is a dual group ring, endowed with a nontrivial associator with the help
of a 3-cocycle, a construction that has also been used frequently, also in the particularly simple case of a group of order 2, as we confirm by giving several explicit references. The fact that this quasi-Hopf algebra leads to the semion datum means that practically all our conjectures for Hopf algebras are false for quasi-Hopf algebras: Besides the fact
that g^{2} ≠ g'^{2} already mentioned, the semion datum is a projective congruence datum that is not a congruence datum in the sense of Definition 2.1. The semion datum also shows that the congruence properties explicitly depend on how the modular datum is extended by the choice of a generalized rank and a multiplicative central charge, and, as mentioned above, it provides an explicit projective representation of a reduced modular group that cannot be lifted to a linear representation, thus confirming the nontriviality of the corresponding Schur multiplicator.

Finally, let us say some words about the conventions that we use throughout the exposition. Our base field is denoted by K, and while we always assume that it is of characteristic zero, we do not always assume that it is algebraically closed. The dual of a vector space V is denoted by V^{*}:=Hom_{K}(V,K). Unless stated otherwise, a module is a left module. If R is a ring, then R^{×} denotes its group of units. Also, we
use the so-called Kronecker symbol δ_{ij}, which is equal to 1 if i=j and equal to 0 otherwise. The set of natural numbers is the set N:={1,2,3,...}; in particular, 0 is not a natural number. The symbol Q_{N} denotes the N-th cyclotomic field, and not some field of N-adic numbers, and Z_{N} denotes the set Z/NZ of integers modulo N, and not some ring of N-adic integers.
The greatest common divisor of two integers k and l is denoted by gcd(k,l) and is always chosen to be nonnegative. With respect to enumeration, we use the convention that propositions, definitions, and similar items are referenced by the paragraph in which they occur; an
additional third digit indicates a part of the corresponding item.
For example, a reference to Proposition 1.1.1 refers to the first
assertion of the unique proposition in Paragraph 1.1.

Part of the present work was carried out during a visit of the first author to the Hong Kong University of Science and Technology. He thanks the university, and in particular his host, for the hospitality.

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