اشترك بالحزمة الذهبية واحصل على وصول غير محدود شمرا أكاديميا
تسجيل مستخدم جديدGeneralized and quasi-localizations of braid group representations
138
0
0.0
(
0
)
اسأل ChatGPT حول البحث
ﻻ يوجد ملخص باللغة العربية
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.
قيم البحث
اقرأ أيضاً
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 ar
e 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.
سجل دخول لتتمكن من نشر تعليقات
التعليقات
جاري جلب التعليقات
سجل دخول لتتمكن من متابعة معايير البحث التي قمت باختيارها
