Stack Exchange Network

Stack Exchange network consists of 175 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange

Questions tagged [fusion-categories]

The tag has no usage guidance.

1
vote
0answers
57 views

Finding the Drinfeld centre of a category

I have the following unitary monoidal spherical category C: Simple objects: $\{1,x,y\}$. Non-trivial Fusion Rules: $$x\otimes y=x=y\otimes x$$ $$x\otimes x=1 \oplus 2x \oplus y$$ I would like to ...
1
vote
0answers
91 views

Multi-sensor fusion with full data sets

I have noisy position tracks of a moving object from multiple sensors, and I want to fuse these tracks to come up with a single "best" estimate of the object's position. Unlike in most applications, I ...
1
vote
1answer
42 views

Is the twist of the direct sum of simple objects the direct sum of the twists?

$\DeclareMathOperator{\id}{id} \DeclareMathOperator{\tr}{tr}$I am reading Bakalov and Kirillov's Lectures on tensor categories and modular functors, and based on notation from section $2.4$, and ...
0
votes
1answer
93 views

$G$-Graded vector spaces and Yetter cohomology

In this article, the following claims are made: The Yetter cohomology of $G$-graded $k$-vector spaces, for $G$ a finite group and $k$ algebraically closed, is given by group cohomology of $G$ with ...
5
votes
1answer
199 views

Example of non-isomorphic, Morita-equivalent semisimple Hopf-algebras

In the paper http://arxiv.org/abs/1509.01548, section 1.3, I found the following definitions: Two fusion categories $\mathcal{C}$ and $\mathcal{D}$ are Morita-equivalent if there exists an ...
2
votes
0answers
132 views

Half-Twists in Ribbon Categories

I've been doing some reading on ribbon categories, and something that's caught my attention lately is the ribbon category with half twist, see for example here. From my understanding, the objects in ...
2
votes
1answer
206 views

Introduction to subfactor theory

I have almost no knowledge about subfactor theory but I would like to understand what it is. As a self-learner, I do not know where to start. Could you suggest introductory text/paper/book to study ...
3
votes
0answers
62 views

How to calculate braiding eigenvalues in a fusion category?

Statements like this are found in published articles: The context: Assume $\mathcal{C}$ is a complex fusion category (i.e. complex linear, finitely semisimple, monoidal, with duals, with simple ...
3
votes
0answers
63 views

How to construct a G-extension of a category C?

Note: I'm a physicist so this will be phrased somewhat in physics language. Suppose we have a unitary modular tensor category $\mathcal{C}$. In physics language, we can think of $\mathcal{C}$ as ...
3
votes
0answers
200 views

Frobenius-Perron dimension on a fusion category

Let $C$ be a fusion category with simple objects $V_i\in I$. The fusion rule is $V_i\otimes V_j \cong N_{i,j}^k V_k$. The Frobenius-Perron dimension of a simple object $V_i$, $\mathrm{FPdim}(i)$, is ...
2
votes
0answers
181 views

representation of a group and its center

Let $G$ be a finite group and let $Z(G)$ be its center. Let $C=\mathrm{Rep}(G)$ be the category of finite dimensional representation of $G$. Let $D$ be the fusion subcategory of $C$ generated by $V \...
6
votes
1answer
216 views

Is Tambara-Yamagami category admits a braiding when G is a nonabelian group?

Tambara-Yamagami category is a fusion category which its simple objects are elements of a group and one element out of group. i.e : $$simple\;objects = G \cup \{m\}$$ The fusion rule of this ...
2
votes
1answer
261 views

When is an object in a linear or abelian category simple? Or: How should I define fusion categories?

I'm confused about the notion of simple objects. Now ncatlab says that an object is simple in an abelian category if it only has itself and 0 as subobjects. On another page, it says that the simple ...