Modular categories of dimension 𝑝³𝑚 with 𝑚 square-free

The fusion rules in [Formula: see text] for a finite group [Formula: see text] can be computed in terms of character inner products. Using an explicit formula (Theorem 3.2) for these fusion rules, we show that [Formula: see text] is multiplicity free for two infinite families of finite groups: the Dihedral groups and the Dicyclic groups. In fact, we will compute all fusion rules in these categories. Multiplicity freeness is a desired property for modular tensor categories, since it greatly simplifies the computation of [Formula: see text]-matrices. Furthermore, we observe that the fusion rules for Dihedral groups [Formula: see text] with [Formula: see text] odd are extremely similar to the fusion rules of type-B level-[Formula: see text] fusion algebras of Wess–Zumino–Witten conformal field theories. Moreover, we give a proof of the fusion rule formula by using Mackey theory.

In this paper, we study modular categories whose Galois group actions on their simple objects are transitive. We show that such modular categories admit unique factorization into prime transitive factors. The representations of $SL_2(\mathbb{Z})$ associated with transitive modular categories are proven to be minimal and irreducible. Together with the Verlinde formula, we characterize prime transitive modular categories as the Galois conjugates of the adjoint subcategory of the quantum group modular category $\mathcal{C}(\mathfrak{sl}_2,p-2)$ for some prime $p > 3$. As a consequence, we completely classify transitive modular categories. Transitivity of super-modular categories can be similarly defined. A unique factorization of any transitive super-modular category into s-simple transitive factors is obtained, and the split transitive super-modular categories are completely classified.

An abelian k-linear semisimple category having a finite number of simple objects, and endowed with a ribbon structure, is called premodular. It is modular (in the sense of Turaev) if the so-called S-matrix is invertible. A modular category defines invariants of 3-manifolds and a TQFT ([T]). When is it possible to construct a modularisation of a given premodular category, i.e. a functor to a modular category preserving the structures and ‘dominant’ in a certain sense? It turns out (2.3) that this amounts essentially to making ‘transparent’ objects trivial. We give a full answer to this problem in the case when k is a field of char. 0 (as well as partial answers in char. p): under a few obvious hypotheses, a premodular category admits a modularisation, which is unique (th. 3.1, and cor. 3.5 in char. 0) The proof relies on two main ingredients: a new and very simple criterion for the S-matrix to be invertible (1.1) and Deligne's internal characterization of tannakian categories in char. 0 [D]. When simple transparent objects are invertible, the criterion is simpler (4.2) and the modularisation can be described more explicitly (prop. 4.4). We conclude with two examples: the premodular categories associated with quantum $\hbox{\sf SL}_N$ and $\hbox{\sf PGL}_N$ at roots of unity; in the first case, we obtain modular categories which were built independently by C. Blanchet [B]; in the second case, we obtain modularizations in all the cases where Y. Yokota [Y] found Reshetikhin-Turaev invariants of 3-manifolds, thereby improving as well as explaining Yokota's results.

The hypothesis in Theorem 40 that F contain certain roots of unity can be dropped, but we give a preliminary discussion from group theory before proving this.

The aim of this chapter is to determine a normal element for every extension E/F of finite fields. For this, it will be convenient to work (theoretically) in an algebraic closure \(\widehat{F}\) of a fixed ground field F = GF(q) with q elements. While this will simplify the presentation, it is not actually required for constructing normal elements in specific cases: then all that is really needed are finite extensions of F. Recall from Chap. 4 that for every \(n \in {\mathbb{N}}^{*}\) there is exactly one intermediate field En of \(\widehat{F}/F\) with degree n over F. We will explicitly describe a normal element wn for the extension En/F in terms of certain roots of unity whose multiplicative orders depend on q and n. Moreover, we give an inductive construction for a trace-compatible sequence \((w_{n} )_{{n \in {\mathbb{N}}^{*} }}\) over \(\widehat{F}\) for which every wn is a normal element for En/F. Finally, we also present two reasonably efficient algorithms for computing a normal element for an arbitrary extension E = GF(qn) of F = GF(q).

It is shown how to deduce integrality properties of quantum 3-manifold invariants from the existence of integral subcategories of modular categories. The method is illustrated in the case of the invariants associated to classical Lie algebras constructed in [42], showing that the invariants are algebraic integers provided the root of unity has prime order. This generalizes a result of [31], [32] and [29] in the sl2-case. We also discuss some details in the construction of invariants of 3-manifolds, such as the S-matrix in the PSUk case, and a local orientation reversal principle for the colored Homfly polynomial.

We describe a systematic way of producing fermionic topological phases using the technique of fermion condensation. We give a prescription for performing fermion condensation in bosonic topological phases which contain an emergent fermion. Our approach to fermion condensation can roughly be understood as coupling the parent bosonic topological phase to a phase of physical fermions, and condensing pairs of physical and emergent fermions. There are two distinct types of objects in fermionic theories, which we call “m-type” and “q-type” particles. The endomorphism algebras of q-type particles are complex Clifford algebras, and they have no analogues in bosonic theories. We construct a fermionic generalization of the tube category, which allows us to compute the quasiparticle excitations in fermionic topological phases. We then prove a series of results relating data in condensed theories to data in their parent theories; for example, if C is a modular tensor category containing a fermion, then the tube category of the condensed theory satisfies Tube(C/ψ) ≅ C × C/ψ. We also study how modular transformations, fusion rules, and coherence relations are modified in the fermionic setting, prove a fermionic version of the Verlinde dimension formula, construct a commuting projector lattice Hamiltonian for fermionic theories, and write down a fermionic version of the Turaev-Viro-Barrett-Westbury state sum.

This is the second part of the paper (the first part is published in Jour. of AMS, vol.9, 1135--1170, q-alg/9508017). In the first part, we defined for every modular tensor category (MTC) inner products on the spaces of morphisms and proved that the inner product on the space $\Hom (\bigoplus X_i\otimes X^*_i, U)$ is modular invariant. Also, we have shown that in the case of the MTC arising from the representations of the quantum group $U_q \sln$ at roots of unity and $U$ being a symmetric power of the fundamental representation, this inner product coincides with so-called Macdonald's inner product on symmetric polynomials. In this paper, we apply the same construction to the MTC coming from the integrable representations of affine Lie algebras. In this case our construction immediately gives a hermitian form on the spaces of conformal blocks, and this form is modular invariant (Warning: we cannot prove that it is positive definite). We show that this form can be rewritten in terms of asymptotics of KZ equations, and calculate it for $sl_2$, in which case the formula is a natural affine analogue of Macdonald's inner product identities. We also formulate as a conjecture similar formula for $sl_n$.

We present an algorithm for computing Gauss sums over Fq for large prime powers q. This allows us to find the exact values of Gauss sums in many previously intractable cases. The efficient computation of such Gauss sums up to multiplication with a root of unity is achieved by using Stickelberger’s factorization of Gauss sums, an application of the Fincke-Pohst algorithm, and fast arithmetic for subfields of cyclotomic fields. This still leaves the crucial problem of resolving the root of unity ambiguity, which we settle by recursively computing “H-polynomials”. These are integer polynomials which interpolate Gauss sums at certain roots of unity. As an application, we compute the weight distribution of all binary irreducible cyclic codes of length n and dimension k with 2 k−1 n ≤ 5, 000. 2000 Mathematics Subject Classification: 11T24 (primary), 11Y40 (secondary)

Let V be a simple vertex operator algebra satisfying the following conditions: (i) V(n)) = 0 for n < 0, V(0)=C1, and the contragredient module V' is isomorphic to V as a V-module; (ii) every N-gradable weak V-module is completely reducible; (iii) V is C(2)-cofinite. We announce a proof of the Verlinde conjecture for V, that is, of the statement that the matrices formed by the fusion rules among irreducible V-modules are diagonalized by the matrix given by the action of the modular transformation tau |--> -1/tau on the space of characters of irreducible V-modules. We discuss some consequences of the Verlinde conjecture, including the Verlinde formula for the fusion rules, a formula for the matrix given by the action of tau |--> -1/tau, and the symmetry of this matrix. We also announce a proof of the rigidity and nondegeneracy property of the braided tensor category structure on the category of V-modules when V satisfies in addition the condition that irreducible V-modules not equivalent to V have no nonzero elements of weight 0. In particular, the category of V-modules has a natural structure of modular tensor category.

We suggest a new construction for the Quantum Groups - Jones polynomials of torus knots in terms of the PBW theorem of DAHA for any root systems and weights (justified for type A). The main focus is on the DAHA super-polynomials, a stable 3-parametric type A variant of this construction. A connection is expected with the approach to super-polynomials due to Aganagic and Shakirov via the Macdonald polynomials at roots of unity and the Verlinde algebra. The duality conjecture for the DAHA super-polynomials is stated, essentially matching that due to Gukov and Stosic. A link to Khovanov-Rozansky polynomials is provided, including small N (for some torus knots). The hyper-polynomials of types B and C are defined, generalizing the Kauffman invariants and containing an extra parameter vs. the super-polynomials. The special values and other features of the DAHA super and hyper-polynomials are discussed; there are many examples in the paper.

Antiholomorphic Representations for Orthogonal and Symplectic Quantum Groups

We consider some conditions similar to Ozawa’s condition (AO) and prove that if a non-injective factor satisfies such a condition and has the W ∗ C B A P \rm W^*CBAP , then it has no Cartan subalgebras. As a corollary, we prove that I I 1 \rm II_1 factors of universal orthogonal and unitary discrete quantum groups have no Cartan subalgebras. We also prove that continuous cores of type I I I 1 \rm III_1 factors with such a condition are semisolid as a I I ∞ \rm II_\infty factor.

We introduce dynamical analogues of the free orthogonal and free unitary quantum groups, which are no longer Hopf algebras but Hopf algebroids or quantum groupoids. These objects are constructed on the purely algebraic level and on the level of universal C*-algebras. As an example, we recover the dynamical SU_q(2) studied by Koelink and Rosengren, and construct a refinement that includes several interesting limit cases.

Introduction. In [Jon83], Jones constructs, from any III subfactor pair M,N with finite index [M, N] = q + q-I + 2, a canonical hyperfinite subfactor pair of the same index. The pair is constructed from the completion of the TemperleyLieb algebra by a Markov trace. By showing that such a completion yields a II, subfactor pair exactly when q is a positive real or of the form e27ri/k, he obtained his index of subfactors result. The fact that the set of admissible values of q divides into a discrete piece and a continuous piece is puzzling and is indicative of qualitative differences between the subfactors themselves. For one thing, at roots of unity the subfactors are of finite depth. That is, after a certain point in the Bratteli diagram the centers of the finite-dimensional approximating algebras are periodically isomorphic. On the contrary, in the positive real case each finite-dimensional algebra in the Bratteli diagram has a strictly larger center than the previous one. Furthermore, the relative commutant of the smaller algebra in the larger algebra consists only of multiples of the identity in the root of unity case. For positive real q, not only is there a relative commutant, but in fact the subfactor is locally trivial. That is, the subfactor pair, restricted to a minimal projection in the relative commutant, is the trivial subfactor pair. In addition to its direct relevance to the study of subfactors, Jones' work had indirect but important ramifications. The Temperley-Lieb algebra in the construction forms a representation of the braid group, and the Markov trace as a functional on the braid group is, with proper normalization, a polynomial link invariant. This discovery triggered an explosion of mathematics which eventually came to see the algebra, trace, and invariant as a special case of a general construction which associates, via quantum groups, an example of all three to each irreducible representation of each. simple Lie algebra. The original Jones polynomial comes from the fundamental representation of s12. The quantum groups construction can be done entirely without reference to subfactors, or even analysis. Nevertheless, there are echos of Jones' work, and in particular both the roots of unity and the positive real q continue to play a distinct and special role. For example, when q is a root of unity, the quantum group