From quantum groups to unitary modular tensor categories

Abstract

Abstract. Modular tensor categories are generalizations of the representation categories of quantum groups at roots of unity axiomatizing the properties necessary to produce 3-dimensional TQFTs. Although other constructions have since been found, quantum groups remain the most prolific source. Recently proposed applications to quantum computing have provided an impetus to understand and describe these examples as explicitly as possible, especially those that are "physically feasible." We survey the current status of the problem of producing unitary modular tensor categories from quantum groups, emphasizing explicit computations.

Unitarizability of premodular categories

“…We make no attempt to be entirely self-contained, referring the reader to the survey paper [13] and the texts [2,17] for full details.…”

“…The category C(g 2 , q, 7) is the trivial rank 1 category for any choice of q, and the rank of C(g 2 , q, ) for 3 can be computed via the generating function [13]. If we label the short fundamental weight (corresponding the the seven-dimensional rep of g 2 ) by Λ 1 and the long fundamental weight by Λ 2 we obtain the following formula for dim q of the simple object X λ labeled by weight λ = λ 1 Λ 1 + λ 2 Λ 2 : .…”

“…. as can be obtained from the generating function [13]). We label the four fundamental weights by Λ i , 1 ≤ i ≤ 4 where Λ 1 corresponds to the 26-dimensional representation of f 4 .…”

On Classification of Modular Tensor Categories

“…No Galois actions can change theS matrix, hence the quantum dimension of the non-trivial simple object from −1 to 1, though the same fusion rules can be realized by a unitary theory: the semion theory. Reference [Ro1] contains a set of fusion rules which has non-unitary MTC realizations, but has no unitary realizations at all. …”

Topological-Like Features in Diagrammatical Quantum Circuits

“…But for dagger ribbon categories with some specific fusion rules, one can not achieve both positive definite forms and unitary braidings due to compatibility conditions between them, see interesting examples in Rowell's papers [42,43]. The manuscript [44] to be an extension of the proposal in the first footnote and last section will completely exploit Turaev's notations on Hermitian ribbon categories.…”