On a family of non-unitarizable ribbon categories

Abstract

We consider several families of categories. The first are quotients of H. Andersen's tilting module categories for quantum groups of Lie type B at odd roots of unity. The second consists of categories of type BC constructed from idempotents in BMW -algebras. Our main result is to show that these families coincide as braided tensor categories using a recent theorem of Tuba and Wenzl. By appealing to similar results of Blanchet and Beliakova we obtain another interesting equivalence with these two families of categories and the quantum group categories of Lie type C at odd roots of unity. The morphism spaces in these categories can be equipped with a Hermitian form, and we are able to show that these categories are never unitary, and no braided tensor category sharing the Grothendieck semiring common to these families is unitarizable.

Unitarizability of premodular categories

“…Kirillov Jr. [9] defined an Hermitian structure on C(g, q, ), and conjectured that for the choice q = e πi/ the form is positive definite provided 2 | if g is of Lie type B, C or F 4 and 3 | if g is of Lie type G 2 . Subsequently, Wenzl [18] proved this conjecture, and we showed [12] that the hypothesis 2 | is necessary for Lie types B and C for sufficiently large. In fact in [12] it is shown that unitarity fails in a much stronger sense: no Hermitian premodular category with the same Grothendieck semiring as C(so 2k+1 , q, ) can even be pseudo-unitary when is odd with 4k + 3 ≤ .…”

“…Subsequently, Wenzl [18] proved this conjecture, and we showed [12] that the hypothesis 2 | is necessary for Lie types B and C for sufficiently large. In fact in [12] it is shown that unitarity fails in a much stronger sense: no Hermitian premodular category with the same Grothendieck semiring as C(so 2k+1 , q, ) can even be pseudo-unitary when is odd with 4k + 3 ≤ . This result was obtained by appealing to the classification results found in [16].…”

Singly generated planar algebras of small dimension, Part III

“…It is worth mentioning that the second-to-last row in Table 1 should be excluded. This is clarified in [Row05], and Theorem 3.8 in [Row08]. When q ≥ 1, the argument is similar to Corollary 5.6 in [Wen90].…”

Two paradigms for topological quantum computation

“…These are obtained as subquotients of the category of finite dimensional representations of the quantum group U q g, see [35] for a survey. Such a category may fail to be modular or unitary (see [34] and [36]), but such circumstances can be avoided by certain restrictions on ℓ. Specifically, define m = 1 for Lie types A, D and E, m = 2 for Lie types B, C and F 4 and m = 3 for Lie type G 2 .…”