Unitarizability of premodular categories

Abstract

We study the unitarizability of premodular categories constructed from representations of quantum group at roots of unity. We introduce Grothendieck unitarizability as a natural generalization of unitarizability to classes of premodular categories with a common Grothendieck semiring. We obtain new results for quantum groups of Lie types F 4 and G 2 , and improve the previously obtained results for Lie types B and C.

Knot polynomial identities and quantum group coincidences

“…These conceptual explanations do not suffice for the equivalence between the even part of D 14 and Rep U exp(2πi 26 ) (g 2 ), = −3 or 10, which deserves further exploration. Nonetheless we can prove this equivalence using direct methods (see Section 3.5), and it answers a conjecture of Rowell's [36] concerning the unitarity of (G 2 ) 1 3 .…”

“…By §3.4 and Theorems 3.1 and 3.2 we have a functor from the Dubrovnik category with a = exp(2πi 1 13 ) and z = exp(2πi 1 26 ) − exp(2πi −1 26 ) to Rep U exp(2πi 26 ) (g 2 ). Since the target category is pseudo-unitary [36], this functor factors through the semisimplification of the diagram category, which is the premodular category Rep U q=exp(2πi 1 52 ) (so (3)). Since the target is modular [38] and the functor is dominant (a straightforward calculation via the Racah rule in the Grothendieck group of Rep U exp(2πi 26 ) (g 2 )) this functor induces an equivalence between the modularization of Rep U q=exp(2πi 1 52 ) (so (3)), which is nothing but 1 2 D 14 , and Rep U exp(2πi 26 ) (g 2 ).…”

The little desert? Some subfactors with index in the interval $(5, 3+\sqrt{5})$

“…Given a quantum group U q (g), irreducible representation V and root of unity ζ = exp(2πi/ ), there is a corresponding subfactor Q(g, V, ), as long as a certain positivity condition is satisfied [Wen98]. (This condition has been completely analysed in [Wen90,Row05,Row08,MPS11].) We are interested in the subfactors A = Q(su 2 , V (2) , 14) and B = Q(su 3 , V (1,0) , 14).…”

Quantum Computation and Real Multiplication

“…The vacuum 1 is self-dual. The fusion coefficients satisfy the identities [30]. Let Λ i = diag(λ ij ) be the diagonal matrix with entries the eigenvalues λ ij of the matrix N i , and let S i be the matrix with columns the corresponding eigenvectors.…”