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.
The geometric meaning of quantum Gauss decomposition, manuscript 2020 (replacing 2003 manuscripts Localized coinvariants I,II)
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,
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. I had partial results in this direction in 2004, when I wrote a LaTeX manuscript “Globalizing Hopf-Galois extensions”. Several main remaining problems were resolved in collaboration with Gabi in Spring 2011 when we wrote few new manuscripts. It will soon appear as joint work with the same title
There is some complementary work on gluing functors which may stay part of this work or will appear elsewhere (in a bit different setup the topic of gluing functors has been recently continued in a collaboration with M. Stojić).
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
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 309:1, 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 universal enveloping algebras I, 14 pp. pdf arXiv:0806.0978 (radical revision v2, June 2020; accepted v3 to be posted) accepted in Proc. Conf. Representation Theory XVI, Dubrovnik, IUC, June 19-25, 2019 (Contemporary Math. 2000.)
S. Meljanac, Z. Škoda, Leibniz rules for enveloping algebras and a diagrammatic expansion, pdf
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)
A note on symmetric orderings, to appear in Acta Math. Spalatiensia (AMAS), Ser. Sci. (2020) arxiv/2001.10463
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
and the discussion of algebroid version of Drinfeld twists producing such algebroids out of an algebroid of differential operators is in
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, Int. J. Mod. Phys. A 35:8, 2050034, 15pp. arxiv/1911.03967
D. Meljanac, S. Meljanac, Z. Škoda, R. Štrajn, On interpolations between Jordanian twists, arxiv/2003.01036
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 $H\otimes H$ of the total algebra $H$ 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 $\mathcal{B}$:
In October 2019, at IHES I proved that if $H$ is a Hopf $A$-algebroid in the sense of Bohm-Szlachanyi and $F$ a Drinfeld-Xu counital bialgebroid 2-cocycle, then the bialgebroid twist of $H$ has an antipode. Showing this has been elusive since 1999 definition of the bialgebroid twist by Xu. I am now writing an account of this result. hopefully to be out by the end of February or early March,
Integrable models
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:
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
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.
Popularization
Last revised on June 19, 2020 at 16:28:09. See the history of this page for a list of all contributions to it.