q-Index on braided non-commutative spheres

To some Yang-Baxter braidings of Hecke type we assign algebras called braided non-commutative spheres. For any such algebra, we introduce and compute a q-analog of the standard pairing Ind :K0(A)×K0(A)→Z called a non-commutative index. Unlike the standard non-commutative index, our q-analog is based on the so-called categorical trace specific for a braided category in which the algebra in question is represented.