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# Niels Kowalzig

INdAM-COFUND Incoming fellow between 2012-06-18 and 2014-06-17

### Project information

#### Title

Algebroids: Hopf algebroids, quantum groupoids, and applications

#### Partner organisation

Università degli Studi di Roma “Tor Vergata”

Via della Ricerca Scientifica, 1

00133 Roma, Italy

#### Abstract

The aim of the proposed research project is to establish the notion of Hopf algebroid (quantum groupoid) as a

fundamental concept of mathematics and mathematical physics by highlighting its ubiquity in various fields, such as algebra, group theory, differential geometry, noncommutative geometry, and mathematical physics.

This may be exemplified by the fact that Hopf algebroids provide a natural framework for unifying and extending classical constructions in homological algebra: Group, groupoid, Lie algebra, Lie algebroid and Poisson (co)homology (hence the relevant data for deformation quantisation), Hochschild and cyclic homology for associative algebras, twisted cyclic homology, as well as Hopf-cyclic homology for Hopf algebras, are all special cases of the cyclic homology of Hopf algebroids since the rings over which these theories can be expressed as derived functors are all Hopf algebroids. That is to say, Hopf algebroids appear, for example, as the right noncommutative generalisation of Lie algebroids and groupoids.

In addition, higher algebraic structures on these (co)homology groups, such as Gerstenhaber and Batalin-Vilkovisky algebra and module structures, find their natural description in the Hopf algebroid framework. These structures are also relevant in yet different fields, such as quantum field theory and deformation quantisation, which should eventually lead to a formality conjecture for Hopf algebroids.

On our way we will investigate all kinds of ramifications of the theory, such as functoriality, an equivariant theory, Morita equivalence and invariance, generalised Connes-Moscovici algebras (i.e., bicrossed products) for G-structures and pseudogroups. Operads for categories of non-symmetric bimodules will provide a language to handle the technical details in a manageable size.

### Fellow information

#### Research interests

- Noncommutative geometry, homological algebra, cyclic homology;
- Hopf algebr[a/oid]s, quantum group[oid]s;
- Lie-Rinehart algebras, (étale) groupoids;
- Poisson geometry, (deformation) quantisation, operator algebras.

#### Previous positions, awards

*Jan-May 2011*: Fellow of the Granada Excellence Network (GENIL), Universidad de Granada*DecâSep 2010*: Visiting grant at the Institut des Hautes Etudes Scientifiques, Bures-sur-Yvette, France*June 2009*: PhD in Mathematics, Universiteit Utrecht and Universiteit van Amsterdam*Jan-Apr 2007*: CNRS Marie Curie pre-doc fellowship, Institut Henri Poincaré, Paris*May-Jul 2004*: CNRS Marie Curie pre-doc fellowship, Institut Henri Poincaré, Paris

#### Publications, preprints, other works

- L. El Kaoutit, N. Kowalzig,
*Morita theory for Hopf algebroids, principal bibundles, and weak equivalences*, preprint (2014); arXiv:1407.7461 - S. Chemla, F. Gavarini, N. Kowalzig,
*Duality features of left Hopf algebroids*, preprint (2014); arXiv:1407.7112 - N. Kowalzig,
*Gerstenhaber and Batalin-Vilkovisky structures on modules over operads*, preprint (2013); arXiv:1312.1642, to appear in Int. Math. Res. Not. - N. Kowalzig,
*Batalin-Vilkovisky Algebra Structures on (Co)Tor and Poisson Bialgebroids*, preprint (2013); arXiv:1305.2992, to appear in J. Pure Appl. Algebr. - N. Kowalzig, U. KrÃ€hmer,
*Batalin-Vilkovisky structures on Ext and Tor*, (2012); arXiv:1203.4984, J. Reine Angew. Math. 697 (2014), 159-219 - N. Kowalzig, L. El Kaoutit,
*Morita base change in Hopf-cyclic (co)homology*, (2011); arXiv:1111.3890, Lett. Math. Phys. 103 (2013), no. 6, 665-699 - N. Kowalzig, U. KrÃ€hmer,
*Cyclic structures in algebraic (co)homology theories*, Homology, Homotopy and Applications 13 (2011), no. 1, 297â318; DOI:10.4310/HHA.2011.v13.n1.a11 - N. Kowalzig, H. Posthuma,
*The cyclic theory of Hopf algebroids*, Journal of Noncommutative Geometry 5 (2011), no. 3, 423â476; DOI:10.4171/JNCG/82 - N. Kowalzig, U. KrÃ€hmer,
*Duality and products in algebraic (co)homology theories*, Journal of Algebra 323 (2010), 2063â2081; DOI:10.1016/j.jalgebra.2009.12.026 - N. Kowalzig,
*Hopf algebroids and their cyclic theory*, Ph. D. thesis (2009); available at http://igitur-archive.library.uu.nl/dissertations/2009-0702-200408/UUindex.html - N. Kowalzig, N. Neumaier, M. Pflaum,
*Phase space reduction of star products on cotangent bundles*, Ann. Henri PoincarÃ© 6 (2005), 485â552; DOI:10.1007/s00023-005-0215-y - N. Kowalzig,
*Zur Anwendung der Deformationsquantisierung*, Diploma (master’s) thesis (2001); available at http://www.ihes.fr/~kowalzig/diplkowalzig.ps

#### Conferences, schools, other events

*Incontri di geometria noncommutativa: a Neapolitan Workshop on Noncommutative Geometry*

Naples, Italy, 20-22 September 2012*Future directions for quantum groups*

Lancaster University, 6-08 September 2012*invited talk*

Talk delivered at*workshop Interfaces of Noncommutative Geometry with the Representation Theory*

of Hopf Algebras and Artin Algebras

Istanbul Center for Mathematical Sciences, 7-10 August 2012

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