Ulrich Krähmer

Piecewise principal comodule algebras

A comodule algebra P over a Hopf algebra H with bijective antipode is called principal if the coaction of H is Galois and P is H-equivariantly projective (faithfully flat) over the coaction-invariant subalgebra B. We prove that principality is a piecewise property: given N comodule-algebra surjections P->Pi whose kernels intersect to zero, P is principal if and only if all Pis are principal. Furthermore, assuming the principality of P, we show that the lattice these kernels generate is distributive if and only if so is the lattice obtained by intersection with B. Finally, assuming the above distributivity property, we obtain a flabby sheaf of principal comodule algebras over a certain space that is universal for all such N-families of surjections P->Pi and such that the comodule algebra of global sections is P.

Twisted Homology of Quantum SL(2) - Part II

We complete the calculation of the twisted cyclic homology of the quantised coordinate ring A = Cq[SL(2)] of SL(2) that we began in [14]. In particular, a nontrivial cyclic 3-cocycle is constructed which also has a nontrivial class in Hochschild cohomology and thus should be viewed as a noncommutative geometry analogue of a volume form.

A Lie–Rinehart Algebra with No Antipode

The aim of this note is to communicate a simple example of a Lie–Rinehart algebra whose enveloping algebra is not a Hopf algebroid, neither in the sense of Böhm and Szlachányi, nor in the sense of Lu.

Braided homology of quantum groups

We study braided Hochschild and cyclic homology of ribbon algebras in braided monoidal categories, as introduced by Baez and by Akrami and Majid. We compute this invariant for several examples coming from quantum groups and braided groups.

FRT-duals as quantum enveloping algebras

The Hopf algebra generated by the l-functionals on the quantum double Cq[G]⋈Cq[G] is considered, where Cq[G] is the coordinate algebra of a standard quantum group and q is not a root of unity. It is shown to be isomorphic to Cq[G]op⋈Uq(g). This proves a conjecture by T. Hodges. As an algebra it can be embedded into Uq(g)⊗Uq(g). Here it is proven that there is no bialgebra structure on Uq(g)⊗Uq(g), for which this embedding becomes a homomorphism of bialgebras. In particular, it is not an isomorphism. As a preliminary a lemma of Hodges concerning the structure of l-functionals on Cq[G] is generalized. For the classical groups a certain choice of root vectors is expressed in terms of l-functionals. A formula for their coproduct is derived.

The nodal cubic is a quantum homogeneous space

The cusp was recently shown to admit the structure of a quantum homogeneous space, that is, its coordinate ring B can be embedded as a right coideal subalgebra into a Hopf algebra A such that A is faithfully flat as a B-module. In the present article such a Hopf algebra A is constructed for the coordinate ring B of the nodal cubic, thus further motivating the question which affine varieties are quantum homogeneous spaces.

On the Non-standard Podleś Spheres

It was shown in [1, 5] that the C*-completion of Podleś’ generic quantum spheres A qρ [4] is independent of the parameter ρ. In the present note we provide a proof that this is not true for the A qρ themselves which remained a conjecture in [1]. As a byproduct we obtain that Aut(A qρ) = ℂx