5,467 research outputs found
Constructing spoke subfactors using the jellyfish algorithm
Using Jones' quadratic tangles formulas, we automate the construction of the
4442, 3333, 3311, and 2221 spoke subfactors by finding sets of 1-strand
jellyfish generators. The 4442 spoke subfactor is new, and the 3333, 3311, and
2221 spoke subfactors were previously known.Comment: 44 pages, many figures, to appear in Transactions of the American
Mathematical Societ
Maximising the number of induced cycles in a graph
We determine the maximum number of induced cycles that can be contained in a
graph on vertices, and show that there is a unique graph that
achieves this maximum. This answers a question of Tuza. We also determine the
maximum number of odd or even cycles that can be contained in a graph on vertices and characterise the extremal graphs. This resolves a conjecture
of Chv\'atal and Tuza from 1988.Comment: 36 page
The braid group surjects onto tensor space
Let V be the 7-dimensional irreducible representation of the quantum group
U_q(g_2). For each n, there is a map from the braid group B_n to the
endomorphism algebra of the n-th tensor power of V, given by R-matrices. We can
extend this linearly to a map on the braid group algebra. Lehrer and Zhang
(MR2271576) prove this map is surjective, as a special case of a more general
result.
Using Kuperberg's spider for G_2 from arXiv:math.QA/9201302, we give an
elementary diagrammatic proof of this result.Comment: 9 page
Non-cyclotomic fusion categories
Etingof, Nikshych and Ostrik ask in arXiv:math.QA/0203060 if every fusion
category can be completely defined over a cyclotomic field. We show that this
is not the case: in particular one of the fusion categories coming from the
Haagerup subfactor arXiv:math.OA/9803044 and one coming from the newly
constructed extended Haagerup subfactor arXiv:0909.4099 can not be completely
defined over a cyclotomic field. On the other hand, we show that the double of
the even part of the Haagerup subfactor is completely defined over a cyclotomic
field. We identify the minimal field of definition for each of these fusion
categories, compute the Galois groups, and identify their Galois conjugates.Comment: 22 pages; improved version of Section
Higher categories, colimits, and the blob complex
We summarize our axioms for higher categories, and describe the blob complex.
Fixing an n-category C, the blob complex associates a chain complex B_*(W;C)$
to any n-manifold W. The 0-th homology of this chain complex recovers the usual
topological quantum field theory invariants of W. The higher homology groups
should be viewed as generalizations of Hochschild homology (indeed, when W=S^1
they coincide). The blob complex has a very natural definition in terms of
homotopy colimits along decompositions of the manifold W. We outline the
important properties of the blob complex, and sketch the proof of a
generalization of Deligne's conjecture on Hochschild cohomology and the little
discs operad to higher dimensions.Comment: 7 page
- β¦