Welcome to Zoran's Homepage

Zoran Škoda, Ph.D. University of Wisconsin, Madison, 2002 Associate Professor, University of Zadar and University of Hradec Kralove Orcid: 0000-0002-8415-1732 Email: zskoda@irb.hr summary at Department of Mathematics pdf My nlab pages: bio page, my writings, nlab section, Zoran's scientific career page

Web page of a graduate courses I taught at the University of Zagreb (in Croatian): Sheaves, bundles and cohomology (full acad. yr. 2008/9); Winter 2009/2010: Homotopija u raznim kategorijama (Homotopy in various categories), Geometrija koneksija i integrabilnosti (Geometry of connections and integrability, 2011-2012), Prostori sa strukturnim snopom (Spaces with a structure sheaf, 2012-2013), Homotopy of CW complexes(2014-2015), Descent and nonabelian cohomology (2017-2018).

My main research areas are noncommutative algebraic geometry and geometric aspects of mathematical physics. I have also spent some years on study and projects related to human linguistics (historical and computational) and formal and programming language theory, design, compiler optimization and semantics. Lightly in 2017, and more seriously in 2018 and 2019, I worked on problematics of blockchain.

My thesis was titled COSET SPACES FOR QUANTUM GROUPS (see abstract here ). I used Ore localizations and a new concept of localized coinvariants to construct (without any representation theory, or Ansatze) noncommutative schemes which are in a strict sense coset spaces for matrix quantum groups of type A; the motivation and application was to find the geometric theory (and the appropriate measure) for coherent states for quantum groups. Now I use actions of monoidal categories and various sorts of descent theory to further study similar setups. In 2006 I delved into more categorical algebra and study of various phenomena related to nonabelian cocycles of various sorts; these projects are mainly dormant from about 2012 and will be revisited soon. With C. Schweigert I coordinated a bilateral grant Croatia-Germany on NONABELIAN COHOMOLOGY for 2006-7 and in 2008-2009 I held a small similar bilateral grant with Urs Schreiber also on homological algebra and applications. This grant and hospitality of MPIM Bonn enabled me to participate in the work in progress "twisted nonabelian differential cohomology" , commented recently on category cafe. I am also active in the construction of nlab,a wiki for mathematics and mathematical physics from the point of view of higher category theory and homotopy theory.

My PhD thesis advisor was mathematician J. W. Robbin, and A. Chubukov was my physics supervisor. My research host at Indiana University was Valery Lunts . My boss at IRB was (project 098-0000000-2865) S. Meljanac (he is a theoretical physicist). I participated in a project on homological and geometric methods in rep. theory lead by P. Pandzic at Zagreb university (037-0372794-2807). My basic education is in theoretical physics but in recent years I am more mathematical in my work and interests. Some people with whom I particularly like to talk mathematics are V. A. Lunts, M. Jibladze U. Schreiber, V. Rubtsov and R. Friedrich. I had less chance but it was very useful talking also to T. Maszczyk, G. Sharygin, P. Bressler, B. Noohi, V. Guletskii, G. Bohm, I. Mencattini, Y. Soibelman . In the period 2004-2012 I communicated to a very original researcher in noncommutative algebraic geometry, A. L. Rosenberg, who untimely died in 2012. Recently, I was also influenced by my younger colleague in Zagreb, Igor Bakovic. His thesis is on gerbes and bigroupoid bundles.

I am myself planning a book on "Actions, sheaves and categories", but at this point there is only a preliminary list actionsbk.dvi of parts and chapters.

A partial list of my past conference talks and invited seminar talks may be found here.

You can also download online some of my papers (see also a graphically more attractive list at nlab)

Localizations for construction of quantum coset spaces math.QA/0301090, 34 pages. "Noncommutative geometry and Quantum groups", W.Pusz, P.M. Hajac, eds. Banach Center Publications vol.61, pp. 265--298, Warszawa 2003.

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, ed. A. Ranicki; pp. 220--313, math.QA/0403276.

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

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

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

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

S. Meljanac, Z. Skoda, M. Stojic, Lie algebra type noncommutative phase spaces are Hopf algebroids, Lett. Math. Physics 107:3, 475-503 (2017)

Zoran Skoda, Martina Stojic, Hopf algebroids with balancing subalgebra, J. Alg. 598:445-469 (2022), arXiv:1610.03837 (old version: A two-sided ideal trick in Hopf algebroid axiomatics)

Zoran Skoda, Martina Stojic, Comment on "Twisted bialgebroids versus bialgebroids from Drinfeld twist", arXiv:2308.05083

Zoran Skoda, Martina Stojic, Enveloping algebra is a Yetter--Drinfeld module algebra over Hopf algebra of regular functions on the automorphism group of a Lie algebra, arXiv:2308.15467

Stjepan Meljanac, Zoran Skoda, Sasa Kresic-Juric, Symmetric ordering and Weyl realizations for quantum Minkowski spaces, J. Math. Phys 63 (2022); 123508, 1-15 arXiv:2203.15084

Every quantum minor generates an Ore set , (free-access link) International Math. Res. Notices 2008, rnn063-8; math.QA/0604610

Equivariant monads and equivariant lifts versus a 2-category of distributive laws (arXiv:0707.1609)

(with S. Meljanac) Leibniz rules for enveloping algebras (pdf)

-------- An older, obsolete version is at arXiv: (arXiv:0711.0149)

S. Meljanac, D. Svrtan, Z. Skoda, Exponential formulas and Lie algebra type star products, SIGMA 8 (2012), 013, 1-15, arxiv:1006.0478

H. Sati, U. Schreiber, Z. Skoda, 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

Cech cocycles for quantum principal bundles (arxiv/1111.5316) (now under very essential revision, to appear soon)

Quantum bundles using coactions and localization, preliminary version (2006/7)

A simple algorithm for extending the identities for quantum minors to the multiparametric case (arXiv:0801.4965)

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

Bicategory of entwinings arXiv:0805.4611

Twisted exterior derivative for universal enveloping algebras I , accepted in Proc. Conf. Representation Theory XVI, Dubrovnik, IUC, June 19-25, 2019 (Contemporary Math. 2000.) arXiv:0806.0978

A note on symmetric ordering, Acta Mathematica Spalatensia 1 (2020) 53--60 arXiv:2001.10463

Compatibility of (co)actions and localization (pdf) arXiv:0902.1398 (preliminary version, under revision Feb/March 2020)

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

Heisenberg double versus deformed derivatives, Int. J. Mod. Physics A 27 & 28 (2011) 4845--4854, arXiv:0909.3769.

(with V. Lunts) Hopf modules, {\rm Ext}-groups and descent (incomplete version)

Bi-actegories (May 2007, preliminary version)

(with Gabriella Böhm) Globalizing Hopf-Galois extensions (preliminary version)

Antipodes for Drinfeld-Xu twists of Hopf algebroids (in state of writing, results obtained at IHES in Oct 2019)

Urs Schreiber, Zoran Skoda, Categorified symmetries, Proceedings of 5th Mathematical Physics Meeting, SFIN, XXII Series A: Conferences, No A1, (2009), 397-424 (Editors: Branko Dragovich, Zoran Rakic). Extended 55 page arxiv version: arXiv/1004.2472 (Summer School and Conference on Modern Mathematical Physics, Institute of Physics, Belgrade, Serbia, July 6-17, 2008)

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

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

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

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

S. Meljanac, Z. Skoda, Hopf algebroid twists for deformation quantization of linear Poisson structures, SIGMA 14 (2018) 026, 23 pages (open access), arxiv/1605.01376

D. Meljanac, S. Meljanac, Z. Skoda, R. Strajn, One parameter family of Jordanian twists, SIGMA 15 (2019), 082, 16 pages, arxiv/1904.03993

D. Meljanac, S. Meljanac, Z. Skoda, R. Strajn, Interpolations between Jordanian twists, the Poincare-Weyl algebra and dispersion relations, Int. J. Mod. Phys. A35:8, 2050034, 15 pp. arxiv/1911.03993

D. Meljanac, S. Meljanac, Z. Skoda, R. Strajn, On interpolations between Jordanian twists, Int. J. Mod. Phys. A 35:26, 2050160 (2020) 13pp., arxiv/2003.01036

Prilozi za vrednovanje u teorijskim znanstvenim disciplinama I (34 pages, preliminary version, in Croatian)

Damjan Pistalo is writing an article with me on realization of Lie algebroids by coordinatized vector fields. Damjan had written diploma with me earlier and then finished a PhD thesis on derived D-geometry in Luxembourg under the direction of N. Poncin.

With my ex-student M. Basic, I worked on a programme of finding integration objects for Leibniz algebras, Leibniz groups. Here is a manifesto: Search for Leibniz groups (pdf)

Unfortunately, this interesting program is now dormant already for quite a few years. Recently I started working on a short note about functoriality of actegories induced from (co)module algebras. "Functoriality of actegories from comodule algebras". With Georgiy Sharygin we are working on a project on certain star products related to Lie algebroids. The first phase was to construct, in the presence of a connection, a correction to the symmetrization map for Lie algebroids which would make it compactible with coproduct.