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Title: Module categories and modular invariants
Author: Hardiman, Leonard
ISNI:       0000 0004 8499 4603
Awarding Body: University of Bath
Current Institution: University of Bath
Date of Award: 2019
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Let C be a modular tensor category with a complete set of simples indexed by I. A modular invariant for C is a non-negative integer I × I-matrix that commutes with the modular data of C. In this thesis we present a novel method of associating a non-negative integer I × I-matrix to a pivotal monoidal functor M on C. This is accomplished via a construction called the tube category. The tube category shares all of its objects with C but extends the Hom-spaces. The trace of M naturally extends to a representation of the tube category that we denote TM. As irreducible representations of the tube category are indexed by pairs of elements in I, decomposing TM into irreducibles gives a non-negative integer I × I-matrix, Z(TM). For a general pivotal functor, Z(TM) will not always be a modular invariant; however, it will always commute with the T-matrix. Furthermore, under certain additional conditions on M, it is shown that TM is a haploid, symmetric, commutative Frobenius algebra. Such algebras are known to be connected to modular invariants, in particular a result of Kong and Runkel implies that Z(T M) commutes with the S-matrix if and only if the dimension of T M is equal to the dimension of C. Finally, this procedure is applied to certain pivotal monoidal functors arising from module categories over C.
Supervisor: King, Alastair ; Traustason, Gunnar Sponsor: Not available
Qualification Name: Thesis (Ph.D.) Qualification Level: Doctoral
EThOS ID:  DOI: Not available
Keywords: category theory ; quantum algebra ; tube category