The trace on projective representations of quantum groups

Abstract

Abstract. For certain roots of unity, we consider the categories of weight modules over three quantum groups: small, un-restricted and unrolled. The first main theorem of this paper is to show that there is a modified trace on the projective modules of the first two categories. The second main theorem is to show that category over the unrolled quantum group is ribbon. Partial results related to these theorems were known previously.

Holonomy braidings, biquandles and quantum invariants of links with $$\mathsf {SL} _2(\mathbb {C} )$$ flat connections

“…So (4) implies (1). In the proof of Lemma 3 of [18] it is shown that (1) implies (2) and (2) implies (3). To see (3) implies (4): Let ξ ±ra be the eigenvalues of (−1) +1 ψ(χ).…”

“…This trace restricts to a trace t on the ideal of projective modules Proof. In [17], the use of a Markov trace on the colored braid group was used because at that time the existence of a modified right trace was only known (from [18] we have a modified left and right trace). The above construction is more general because colored braids are Q-tangles and if σ is such a braid whose braid closure is a Q-link L then F (L) = t( F (σ)) by the properties of a modified trace.…”

“…First, in Section 4.2 of [18] it is shown that there is a truncated R-matrix in the h-adic completion which can be specialized at the root of unity ξ to an elementŘ < ∈ U ξ ⊗ U ξ . Since R comes from the conjugation of the h-adic R-matrix and since F r and E r act by zero on V 0 it follows that for any χ ∈ Y we can chooseR in the definition of the braiding so that both c χ 0 ,χ and c χ,χ 0 are given by the action ofŘ < .…”

“…Lemma 6.3 implies that each module V χ in {V χ } χ∈Y is simple and projective. From Corollaries 4 and 5 of[18] it admits a unique (up to global scalar) nontrivial modified trace {t V } V ∈Proj such that for any χ ∈ Y , then there exists α ∈ (C \ Z) ∪ rZ such that χ(Ω) = (−1) r (q α + q −α ) and d(χ) = t Vχ (Id Vχ ) = (−1) r−1 r q (1−r)α + · · · + q (r−3)α + q (r−1)α .…”

The unrolled quantum group inside Lusztig’s quantum group of divided powers

“…Unrolled quantum groups are certain Hopf algebras that are important in quantum topology [GPT09][CGP15] [GP16]. They are used to construct topological invariants from non-semisimple tensor categories, here the representation category of versions of quantum groups at an even root of unity.…”