Zoran Skoda
my writings

Some of my writings can be found online. Below is the list (later I will post here links to some errata and complimentary notes to some of the items). If other authors are not mentioned, the articles are authored just by me (Zoran Škoda). My thesis title was Coset spaces for quantum groups, and the abstract and the other data of it can be read at ProQuest or, in text format, at link. For MR numbers see my MR numbers.

(Co)actions of Hopf algebras, actions of monoidal categories, localizations, descent, quotients

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

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

  • Noncommutative localization in noncommutative geometry, London Math. Society Lecture Note Series 330 (pdf), ed. A. Ranicki; pp. 220–313, math.QA/0403276.

  • Distributive laws for actions of monoidal categories, math.CT/0406310.

  • Included-row exchange principle for quantum minors, math.QA/0510512.

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

  • Equivariant monads and equivariant lifts versus a 2-category of distributive laws, arXiv:0707.1609.

  • A simple algorithm for extending the identities for quantum minors to the multiparametric case, arXiv:0801.4965.

  • Bicategory of entwinings, arXiv:0805.4611.

  • Compatibility of (co)actions and localization, arXiv:0902.1398 (preliminary version).

  • Some equivariant constructions in noncommutative algebraic geometry, Georgian Mathematical Journal 16 (2009), No. 1, 183–202, arXiv:0811.4770; preprint MPIM2009-3.

The following paper is now under serious revision (and a followup is in preparation), please do not take the old arXiv version seriously (it has a number of errors):

There is also a related manuscript on associated bundles,

  • Quantum bundles using coactions and localizations, in preparation.

There is also an ongoing project with Gabi Böhm on comonad compatible localization?s, comonadic machinery for gluing in equivariant setup and on globalizing Hopf-Galois extensions.

Some related entries to the tematics of this group of articles at the nlab are noncommutative geometry, equivariant noncommutative algebraic geometry, noncommutative scheme, quasicoherent sheaf, coherent sheaf, equivariant object. coring, comodule, localized coinvariant, distributive law, algebraic geometry, descent in noncommutative algebraic geometry, noncommutative algebraic geometry, bialgebra, Hopf algebra, Hopf action, Hopf module, comodule algebra, Hopf-Galois extension, quantum group, Ore set, Ore localization, matrix Hopf algebra, quantum Gauss decomposition, universal localization and in ‘private area’ of nlab also gluing categories from localizations, cleft extension of a space cover.

Quantum heaps

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

I introduced quantum heaps in Spring 2001, and they entered as a side-topic (Chapter 9 only) in my thesis. They were rediscovered by Grunspan in 2002 what spurred lots of activity and publications, but unfortunately nobody did refer nor use my work till the publication of my 2007 remake above.

Cyclic homology

Universal enveloping algebras and realizations via formal differential operators

  • N. Durov, S. Meljanac, A. Samsarov, Z. Škoda, A universal formula for representing Lie algebra generators as formal power series with coefficients in the Weyl algebra, Journal of Algebra 309, Issue 1, pp.318-359 (2007) math.RT/0604096, MPIM2006-62.

  • Heisenberg double versus deformed derivatives, Int. J. of Modern Physics A 26, Nos. 27 & 28 (2011) 4845–4854, arXiv:0909.3769, doi.

  • Twisted exterior derivative for enveloping algebras, arXiv:0806.0978.

  • S. Meljanac, Z. Škoda, Leibniz rules for enveloping algebras and a diagrammatic expansion, pdf

    • An older, obsolete version is at arXiv:0711.0149.
  • S. Meljanac, D. Svrtan, Z. Škoda, Exponential formulas and Lie algebra type star products, Symmetry, Integrability and Geometry : Methods and Applications (SIGMA) 8 (2012) , 013; 1-15, arxiv/1006.0478

  • Domagoj Kovačević, Stjepan Meljanac, Andjelo Samsarov, Zoran Škoda, Hermitian realizations of kappa-Minkowski spacetime, arxiv/1307.5772 (hep-th), Int. J. Mod. Phys. A 30 (2015) 1550019, 26 pages doi

  • Jerzy Lukierski, Zoran Škoda, Mariusz Woronowicz, Deformed covariant quantum phase spaces as Hopf algebroids, Physics Letters B 750, 401-406 (2015) arxiv/1507.02612v3 doi

  • Jerzy Lukierski, Zoran Škoda, Mariusz Woronowicz, On Hopf algebroid structure of kappa-deformed Heisenberg algebra, arxiv/1601.01590; Physics of Atomic Nuclei 80:3 (2017) 569–578 (English Version of “Jadernaja Fizika”; conf. volume, ed. G. Pogosyan)

See also related entry symmetric ordering for Lie algebras. We have also exhibited a Hopf algebroid structure on the noncommutative phase space with base coordinate space being a universal enveloping algebra in

  • S. Meljanac, Z. Škoda, M. Stojić, Lie algebra type noncommutative phase spaces are Hopf algebroids, Lett. Math. Phys. 107:3, 475–503 (2017) enhanced pdf (free for online use) doi (Preprint version arxiv/1409.8188 is 32 pp.)

and the discussion of algebroid version of Drinfeld twists producing such algebroids out of an algebroid of differential operators is in

  • S. Meljanac, Z. Škoda, Hopf algebroid twists for deformation quantization of linear Poisson structures, SIGMA 14 (2018) 026; 23 pages arxiv/1605.01376 pdf doi

An example of calculations of the usual Drinfeld twists, this time of Jordanian type is in

  • Daniel Meljanac, Stjepan Meljanac, Zoran Škoda, Rina Štrajn, One parameter family of Jordanian twists, SIGMA 15 (2019), 082, 16 pages, arxiv/1904.03993 doi pdf

  • Daniel Meljanac, Stjepan Meljanac, Zoran Škoda, Rina Štrajn, Interpolations between Jordanian twists, the Poincaré-Weyl algebra and dispersion relations, arxiv/1911.03967

A geometric version of this Hopf algebroid, in terms of differential operators on a Lie group is being developed in an article in preparation with Martina Stojić. We also work on independent construction of the algebraic version of the Hopf algebroid by a construction of extension of scalars for Hopf algebroids generalized to doubles of certain class of filtered infinite-dimensional Hopf algebras.

Meljanac devised in a family of examples an alternative approach to Hopf algebroids over noncommutative base involving some special subalgebra \mathcal{B} of the tensor square HHH\otimes H of the total algebra HH of the Hopf algebroid. In the following work M. Stojić and I have written out an appropriate axiomatics for this kind of constructions; and proven that every scalar extension Hopf algebroid can be cast into this form with an appropriate choice of the balancing subalgebra \mathcal{B}:

  • Zoran Škoda, Martina Stojić, A two-sided ideal trick in Hopf algebroid axiomatics, arxiv/1610.03837

Integrable models

  • Dimitri Gurevich, Vladimir Rubtsov, Pavel Saponov, Zoran Škoda, Generalizations of Poisson structures related to rational Gaudin model, arxiv/1312.7813, Annales Henri Poincaré 16:7, 1689-1707 (2015) doi

Higher stacks, categories and TQFT

  • H. Sati, U. Schreiber, Z. Škoda, D. Stevenson, Twisted nonabelian differential cohomology: Twisted (n-1)-brane n-bundles and their Chern-Simons (n+1)-bundles with characteristic (n+2)-classes, preliminary version, pdf

  • Urs Schreiber, Z. Škoda, Categorified symmetries, 5th Summer School of Modern Mathematical Physics, SFIN, XXII Series A: Conferences, No A1, (2009), 397-424 (Editors: Branko Dragovich, Zoran Rakić). Extended 55 page version: arXiv/1004.2472 (the school/conference took place at the Institute of Physics, Belgrade, Serbia, July 6-17, 2008; SFIN is the abbreviation for series Sveske fizičkih nauka, ISSN 0354-9291).

Other scientific topics

With my student M. Bašić, I work on a programme of finding the integration objects for Leibniz algebras, which could be called Leibniz groups. Here is a manifesto:

  • Search for Leibniz groups (pdf).

Not included in this manifesto is a discussion of the internal Weyl algebra in the Loday-Pirashvili category which is in the rector prize work of Bašić under my guidance. Also there is my manuscript on certain bialgebras in LP category (roughly speaking).

G. Sharygin and I started to write about unfinished research with tentative title

  • G. Sharygin, Z. Škoda, Corrected coexponential map for universal enveloping of a Lie algebroid

The PBW isomorphism is not a coalgebra map in the case of algebroids, that is why the correction. We studied this subject at MPIM Bonn in Spring 2010, then a week at IHES in late November 2010 and talked much via skype in April and should continue these days.

In future, I would like to work closer to the BV-quantization program which is now being better geometrically understood.


  • Portret matematičara Vladimira Arnoljda, Treći program Hrvatskog radija 76 (2010); 169-180 (ISSN0353-9873) (in Croatian; scan will be posted later)

Last revised on December 7, 2019 at 16:55:04. See the history of this page for a list of all contributions to it.