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#### Braid rigidity for path algebras

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2001.11440.pdf

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##### Citation

Martirosyan, L., & Wenzl, H. (in press). Braid rigidity for path algebras.* Indiana University Mathematics Journal,* To appear.

Cite as: https://hdl.handle.net/21.11116/0000-000A-004E-F

##### Abstract

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 $n\in \N$. 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.

powers of an object in a tensor category. If the category is braided, one

obtains representations of the braid groups $B_n$ for all $n\in \N$. 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.