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We give a full classification of all braided semisimple tensor categories whose Grothendieck semiring is the one of Rep À OðyÞ Á (formally), Rep À OðNÞ Á , Rep À SpðNÞ Á or of one of its associated fusion categories. If the braiding is not symmetric, they are completely determined by the eigenvalues of a certain braiding morphism, and we determine precisely which values can occur in the various cases. If the category allows a symmetric braiding, it is essentially determined by the dimension of the object corresponding to the vector representation.H. W. was partially supported by NSF grant DMS 0070848.Brought to you by | Lund University Libraries Authenticated Download Date | 5/25/15 5:11 PM cussions. Imre Tuba would like to thank Ken Goodearl for the same. Both authors would like to thank Alain Bruguières for allowing them to use his unpublished notes [9] in this paper and the referee for the thorough reading and useful remarks which improved the presentation. Definitions and notationWe recall some basic definitions and set up notations. For more details, we refer to [27], [13] for general categorical notions, and to [18], [37] for tensor categories.Definition 2.1. A monoidal category C is a category C with a functor n : C  C ! C called the tensor product, a natural isomorphism a between n ðn  1 C Þ and n ð1 C  nÞ called the associativity constraint, satisfying the pentagon axiom, a unit object 1 A C and natural isomorphisms l X : 1 n X ! X and r X : X n 1 ! X called the left and right unit constraints satisfying the triangle axiom.are 1 X and 1 X à .With this notion of duality, we also have the usual isomorphism between HomðV ; W n X Ã Þ and HomðV n X ; W Þ for any objects V ; W in C. One checks easily that these isomorphisms are given by the maps a ! ð1 W n d X Þ ða n 1 X Þ and Tuba and Wenzl, Braided tensor categories

Metagenomics is a primary tool for the description of microbial and viral communities. The sheer magnitude of the data generated in each metagenome makes identifying key differences in the function and taxonomy between communities difficult to elucidate. Here we discuss the application of seven different data mining and statistical analyses by comparing and contrasting the metabolic functions of 212 microbial metagenomes within and between 10 environments. Not all approaches are appropriate for all questions, and researchers should decide which approach addresses their questions. This work demonstrated the use of each approach: for example, random forests provided a robust and enlightening description of both the clustering of metagenomes and the metabolic processes that were important in separating microbial communities from different environments. All analyses identified that the presence of phage genes within the microbial community was a predictor of whether the microbial community was host-associated or free-living. Several analyses identified the subtle differences that occur with environments, such as those seen in different regions of the marine environment.

We study the problem of deciding whether or not the image of an irreducible representation of the braid group B 3 of degree ≤ 5 has finite image if we are only given the eigenvalues of a generator. We provide a partial algorithm that determines when the images are finite or infinite in all but finitely many cases, and use these results to study examples coming from quantum groups. Our technique uses two classification theorems and the computational group theory package GAP.

Abstract. We characterize all simple unitarizable representations of the braid group B 3 on complex vector spaces of dimension d ≤ 5. In particular, we prove that if σ 1 and σ 2 denote the two generating twists of B 3 , then a simple representation ρ : B 3 → GL(V ) (for dim V ≤ 5) is unitarizable if and only if the eigenvalues λ 1 , λ 2 , . . . , λ d of ρ(σ 1 ) are distinct, satisfy |λ i | = 1 and µ1i are functions of the eigenvalues, explicitly described in this paper.