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Register a new userGeneralized and quasi-localizations of braid group representations
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We develop a theory of localization for braid group representations associated with objects in braided fusion categories and, more generally, to Yang-Baxter operators in monoidal categories. The essential problem is to determine when a family of braid representations can be uniformly modelled upon a tensor power of a fixed vector space in such a way that the braid group generators act locally. Although related to the notion of (quasi-)fiber functors for fusion categories, remarkably, such localizations can exist for representations associated with objects of non-integral dimension. We conjecture that such localizations exist precisely when the object in question has dimension the square-root of an integer and prove several key special cases of the conjecture.
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We give a description of the centralizer algebras for tensor powers of spin objects in the pre-modular categories $SO(N)_2$ (for $N$ odd) and $O(N)_2$ (for $N$ even) in terms of quantum $(n-1)$-tori, via non-standard deformations of $Umathfrak{so}_N$. As a consequence we show that the corresponding braid group representations are Gaussian representations, the images of which are finite groups. This verifies special cases of a conjecture that braid group representations coming from weakly integral braided fusion categories have finite image.
We prove that representations of the braid groups coming from weakly group-theoretical braided fusion categories have finite images.
Path algebras are a convenient way of describing decompositions of tensor powers of an object in a tensor category. If the category is braided, one obtains representations of the braid groups $B_n$ for all $nin N$. We say that such representations are rigid if they are determined by the path algebra and the representations of $B_2$. We show that besides the known classical cases also the braid representations for the path algebra for the 7-dimensional representation of $G_2$ satisfies the rigidity condition, provided $B_3$ generates $End(V^{otimes 3})$. We obtain a complete classification of ribbon tensor categories with the fusion rules of $g(G_2)$ if this condition is satisfied.
A deeper understanding of recent computations of the Brauer group of Hopf algebras is attained by explaining why a direct product decomposition for this group holds and describing the non-interpreted factor occurring in it. For a Hopf algebra $B$ in a braided monoidal category $C$, and under certain assumptions on the braiding (fulfilled if $C$ is symmetric), we construct a sequence for the Brauer group $BM(C;B)$ of $B$-module algebras, generalizing Beatties one. It allows one to prove that $BM(C;B) cong Br(C) times Gal(C;B),$ where $Br(C)$ is the Brauer group of $C$ and $Gal(C;B)$ the group of $B$-Galois objects. We also show that $BM(C;B)$ contains a subgroup isomorphic to $Br(C) times Hc(C;B,I),$ where $Hc(C;B,I)$ is the second Sweedler cohomology group of $B$ with values in the unit object $I$ of $C$. These results are applied to the Brauer group of a quasi-triangular Hopf algebra that is a Radford biproduct $B times H$, where $H$ is a usual Hopf algebra over a field $K$, the Hopf subalgebra generated by the quasi-triangular structure $R$ is contained in $H$ and $B$ is a Hopf algebra in the category ${}_HM$ of left $H$-modules. The Hopf algebras whose Brauer group was recently computed fit this framework. We finally show that $BM(K,H,R) times Hc({}_HM;B,K)$ is a subgroup of the Brauer group $BM(K,B times H,R),$ confirming the suspicion that a certain cohomology group of $B times H$ (second lazy cohomology group was conjectured) embeds into $BM(K,B times H,R).$ New examples of Brauer groups of quasi-triangular Hopf algebras are computed using this sequence.
We establish automorphisms with closed formulas on quasi-split $imath$quantum groups of symmetric Kac-Moody type associated to restricted Weyl groups. The proofs are carried out in the framework of $imath$Hall algebras and reflection functors, thanks to the $imath$Hall algebra realization of $imath$quantum groups in our previous work. Several quantum binomial identities arising along the way are established.
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