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Yayın A characteristic map for compact quantum groups(Springer Heidelberg, 2017-09) Kaygun, Atabey; Sütlü, Serkan SelçukWe show that if G is a compact Lie group and g is its Lie algebra, then there is a map from the Hopf-cyclic cohomology of the quantum enveloping algebra U-q(g) to the twisted cyclic cohomology of quantum group algebra O(G(q)). We also show that the Schmudgen-Wagner index cocycle associated with the volume form of the differential calculus on the standard Podles sphere O(S-q(2)) is in the image of this map.Yayın Homology of quantum linear groups(Int Press Boston, 2021-03-24) Kaygun, Atabey; Sütlü, SerkanFor every n >= 1, we calculate the Hochschild homology of the quantum monoids M-q(n), and the quantum groups GL(q)(n) and SLq(n) with coefficients in a 1-dimensional module coming from a modular pair in involution.Yayın Hopf-cyclic cohomology of quantum enveloping algebras(European Mathematical Society Publishing House, 2016) Kaygun, Atabey; Sütlü, Serkan SelçukIn this paperwe calculate both the periodic and non-periodic Hopf-cyclic cohomology of Drinfeld-Jimbo quantum enveloping algebra Uq.g/ for an arbitrary semi-simple Lie algebra g with coefficients in a modular pair in involution. We obtain this result by showing that the coalgebra Hochschild cohomology of these Hopf algebras are concentrated in a single degree determined by the rank of the Lie algebra g.Yayın The asymptotic Connes-Moscovici characteristic map and the index cocycles(Institute of Mathematics Polish Academy of Sciences, 2020) Kaygun, Atabey; Sütlü, SerkanWe show that the (even and odd) index cocycles for theta-summable Fredholm modules are in the image of the Connes–Moscovici characteristic map. To show this, we first define a new range of asymptotic cohomologies, and then we extend the Connes–Moscovici characteristic map to our setting. The ordinary periodic cyclic cohomology and the entire cyclic cohomology appear as two instances of this setup. We then construct an asymptotic characteristic class, defined independently from the underlying Fredholm module. Paired with the K-theory, the image of this class under the characteristic map yields a non-zero scalar multiple of the index in the even case, and the spectral flow in the odd case.
