Coset spaces for quantum groups

Zoran Škoda, **Coset spaces for quantum groups**, Ph.D. thesis, University of Wisconsin at Madison, defended Jan 17, 2002

Consideration of symmetries often simplifies problems in physics and geometry. Quantum groups are analogues of groups, and they can describe a novel kind of symmetry. We view them as objects of noncommutative geometry. They may act on algebras of noncommutative or quantum observables.

We propose a theory of coset spaces for quantum groups in the language of coactions of Hopf algebras and analyse an example which should be thought as a quantum group analogue of the flag variety. In the classical case, the flag variety is the coset space of the special linear group modulo its Borel subgroup of lower triangular matrices. We introduce and study a notion of localized coinvariants; the quantum group coset space is viewed as a system of algebras of localized coinvariants, equipped with a quantum version of the locally trivial principal bundle where the total space is described by the quantum special linear group and the base space is described by the system of algebras of localized coinvariants.

We use quasideterminants, the commutation relations between the quantum minors and the noncommutative Gauss decomposition to formulate and prove the main results. We apply our axiomatization of quantum group fibre bundles to obtain a generalization of a concept of Perelomov coherent states to the Hopf algebra setting and obtain the corresponding resolution of unity formula.

The content of the thesis within a wider context of noncommutative algebraic geometry based on localization methods is outlined with most of the proofs skipped in

- Localizations for construction of quantum coset spaces, in “Noncommutative geometry and Quantum groups”, W.Pusz, P.M. Hajac, eds. Banach Center Publications vol.61, pp. 265–298, Warszawa 2003, math.QA/0301090.

A crucial lemma on Ore sets generated by quantum minors has been published as

- Every quantum minor generates an Ore set, International Math. Res. Notices 2008, rnn063-8; pdf math.QA/0604610. Note quantum minor Ore erratum.

Still most of the proofs from the rest of the thesis were not published.

Chapter on coherent states has been with written in a more careful exposition

- Coherent states for Hopf algebras, Letters in Mathematical Physics 81, N.1, pp. 1-17, July 2007. (earlier arXiv version: math.QA/0303357)

where an explicit calculation of the measure involved in resolution of unity formula has been added (the latter has been calculated in a 3-daz calculating effort performed at MPIM Bonn.

Chapter introducing quantum heaps is published as

- Quantum heaps, cops and heapy categories , Mathematical Communications 12, No. 1, pp. 1-9 (2007); math.QA/0701749.

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