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 doi

• 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, pdf (older version is 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):

• Čech cocycles for quantum principal bundles, arxiv/1111.5316

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. 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

• Gabriela Böhm, Zoran Škoda, Globalizing Hopf-Galois extensions, in preparation

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

• 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

• Cyclic structures for simplicial objects from comonads, math.CT/0412001.

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, pp. 305–319 in: Lie Groups, Number Theory, and Vertex Algebras, Contemporary Mathematics 768 (2021) (Proc. Conf. Representation Theory XVI, Dubrovnik, IUC, June 19-25, 2019) Edited by: Dražen Adamović, Andrej Dujella, Antun Milas, and Pavle Pandžić; arXiv:0806.0978 doi

• 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)

• Z. Škoda, A note on symmetric orderings, Acta Mathematica Spalatensia 1 (2020) 53–60 pdf arxiv/2001.10463, RBI-ThPhys-2020-36, ISSN 2757-1688 doi

• Stjepan Meljanac, Zoran Škoda, Saša Krešić-Jurić, Symmetric ordering and Weyl realizations for quantum Minkowski spaces, J. Math. Phys. 63 (2022) 123508, 1–15 arXiv/2203.15084 doi

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, 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, Int. J. Mod. Phys. A 35:26, 2050160 (2020) 13pp. arxiv/2003.01036, doi, RBI-ThPhys-2020-37

• S. Meljanac, Z. Škoda, R. Štrajn, Generalized Heisenberg algebra, realizations of the $\mathfrak{gl}(n)$ algebra and applications, Reports on Mathematical Physics 89 (2022) n.1, 131-140 arxiv/2107.03111 doi

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 $\mathcal{B}$:

• Zoran Škoda, Martina Stojić, Hopf algebroids with balancing subalgebra, J. Alg. 598:445-469 (2022) arxiv/1610.03837 doi

A correction and slight generalization of a theorem on twists from a paper of Borowiec and Pachol:

• Zoran Škoda, Martina Stojić, Comment on “Twisted bialgebroids versus bialgebroids from a Drinfeld twist”, J. Phys. A: Math. Theor. 57:10 (2024) 108001 doi 10 pp. arXiv:2308.05083

Another article motivated by scalar extension Hopf algebroids is about an example in

• Zoran Škoda, Martina Stojić, Enveloping algebra is a Yetter–Drinfeld module algebra over Hopf algebra of regular functions on the automorphism group of a Lie algebra, (submitted) arXiv:2308.15467 (v2)

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, currently in draft form.

• Antipodes for Drinfeld-Xu twists of Hopf algebroids

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).

• Athanasios Chatzistavrakidis, Toni Kodžoman, Zoran Škoda, Brane mechanics and gapped Lie $n$-algebroids arXiv:2404.14126, to appear in JHEP

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.

Popularization

I am the author of a radio program text in Croatian, later published as

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

I am one of several coauthors of a mathematical section of a database of Croatian terminology Struna. The official part of mathematical section has been published as

• Goran Igaly, Nenad Antonić, Pavle Goldstein, Ivica Gusić, Miljenko Huzak, Ivana Matas Ivanković, Željka Milin-Šipuš, Siniša Runjaić, Zoran Škoda, Hrvatsko matematičko nazivlje, Goran Igaly editor, I Gusić, I. M. Ivanković, S. Runjaić executive editors, VI+158 pp. Institut za hrvatski jezik i jezikoslovlje, Zagreb, 2015 pdf

Reviewing for MathReviews

I wrote the following reviews for MathSciNet:

• MR4600458 Buchholtz, Ulrik; Weinberger, Jonathan Synthetic fibered $(\infty,1)$-category theory. High. Struct. 7 (2023), no. 1, 74–165. 03B38 (18D30 18D40 18D70 18N45 18N60)

• MR4456599 Pippi, Massimo On the structure of dg categories of relative singularities. High. Struct. 6 (2022), no. 1, 375–402. 14F08 (14A22 14B05 18G80)

• MR4483623 Pronk, Dorette; Scull, Laura Bicategories of fractions revisited: towards small homs and canonical 2-cells. Theory Appl. Categ. 38 (2022), Paper No. 24, 913–1014. 18N10 (18E35)

• MR4413285 Cooney, Nicholas; Grabowski, Jan E. Automorphism groupoids in noncommutative projective geometry. J. Algebra 604 (2022), 296–323. 16S38 (16D90 16W50 20G42)

• MR4328537 Martín Ortiz-Morales, Martha Lizbeth Shaid Sandoval-Miranda, Valente Santiago-Vargas, Gabriel localization in functor categories. Comm. Algebra 49 (2021), no. 12, 5273–5296. 18A25 (16D90 16G10 18E05 18E35)

• MR4325712 Zheng Hua, Guisong Zhou, Quasi-homogeneity of potentials. J. Noncommut. Geom. 15 (2021), no. 2, 399–422. 16S38 (14A22)

• MR4332074 Akhil Mathew, Faithfully flat descent of almost perfect complexes in rigid geometry. J. Pure Appl. Algebra 226 (2022), no. 5, Paper No. 106938, 31 pp. 18F20 (14G22 18G99 18N40 18N60)

• MR4011808 Alexander S. Corner, A universal characterisation of codescent objects. Theory Appl. Categ. 34 (2019), 684–713. 18N10

• MR4112764 M. E. Descotte, E. J. Dubuc, M. A. Szyld, A localization of bicategories via homotopies. Theory Appl. Categ. 35 (2020), No. 23, 845–874. 18N10 (18N40 18N55)

• MR2417986 (2009i:16027) Hiroyuki Minamoto, A noncommutative version of Beilinson’s theorem, J. Algebra 320 (2008), no. 1, 228–252. 16G20 (14F05 16S38 18E30)

• MR2250572 (2007k:16070) P. Jara, L. Merino, G. Navarro, J. F. Ruiz, Localization in coalgebras, stable localizations and path coalgebras, Comm. Algebra 34 (2006), no. 8, 2843–2856. 16W30 (18E35)

• MR2294761 (2008h:16044) Andrzej Tyc, On actions of Hopf algebras on commutative algebras and their invariants, Ann. Univ. Ferrara Sez. VII (N.S.) 51 (2005), 99–103.