Proiectul de cercetare
"Echivalente si dualitati intre categorii de module, aplicatii si generalizari",
Octombrie 2005-Septembrie 2007.
English Version (CEEX_ET19)

Finantat de Autoritatea nationala pentru cercetare stiintifica
Catedra de Algebra, Facultatea de Matematica si Informatica
Universitatea "Babes-Bolyai", Cluj-Napoca
Echipa:

Simion Breaz (director de proiect),
Septimiu Crivei,
Cosmin Pelea,
Szanto Csaba,
Camelia Dicu.
Obiective:

(I) Obiective stiintifice

1) Caracterizari ale grupurilor (abeliene) auto-mici care au proprietati de substitutie sau simplificare.
2) Stabilirea de dualitati intre categorii de module. Se vor construi dualitati care acopera si alte cazuri decat cele acoperite de dualitatile induse de modulele costar, introduse de Colby si Fuller.
3) Descrierea unor categorii de submodule folosind tehnicile initiate de Ringel si Arnold pentru categorii de subgrupuri.
4) Stabilirea de caracterizari ale grupurilor (modulelor) auto-mici cu diverse proprietati de scindare. Vor fi studiate proprietatile de scindare Baer si Szele si modul in care pot fi caracterizate cu ajutorul echivalentelor si dualitatilor.
5) Caracterizari ale unor echivalente si dualitati intre categorii de functori. Se vor generaliza teoremele de tip Morita, Gruson-Jensen, Gabriel-Popescu pentru diferite categorii aditive, categorii de fractii ale unor categorii de functori.
6) Determinarea de legaturi intre existenta unor dualitati, categorii de submodule, generalizari ale algebrelor universale endoprimale si proprietati ale functiilor omogene.

(II) Cresterea vizibilitatii

1. Publicarea rezultatelor obtinute in reviste cu vizibilitate internationala buna (indexate ISI si/sau recenzate in Mathematical Reviews sau Zentralblatt fur Mathematik. 2. Participarea la seminarii de cercetare si la conferinte internationale in care sa fie expuse rezultate ale cercetarilor membrilor echipei.

(III) Imbunatatirea conditiilor de cercetare

1. Completarea infrastructurii de cercetare prin
-- achizitionarea de materiale bibliografice (tratate stiintifice)
-- asigurarea mijloacelor materiale necesare cercetarii (echipamente de birotica, consumabile etc.)
2. Imbunatatirea gradului de informare asupra directiilor de cercetare actuale ale membrilor echipei si ale altor tineri (masteranzi, doctoranzi);
Realizari:
(A) Articole in reviste ISI

1. Cancellation Properties for Quotient Divisible Groups, de U. Albrecht (Auburn University), S. Breaz, C. Vinsohaler (University of Connecticut) si W. Wickless (University of Connecticut), J. Algebra, doi:10.1016/j.jalgebra.2007.03.014; .
2. Source modules of blocks with normal defect groups, Camelia Dicu si A. Marcus, Arch. Math., 88 (2007), 289-296. doi:10.1007/s00013-006-1964-7;
3. Finitely $A$-cogenerated abelian groups, de U. Albrecht (Auburn University), S. Breaz si W. Wickless (University of Connecticut), acceptata de Houston Journal of Mathematics, editor: Jutta Hausen.
4. Every Abelian group is determined by a subgroup lattice, S. Breaz si Gr. Calugareanu, Studia Scientiarum Mathematicarum Hungaricae, doi:10.1556/SScMath.2007.1046; editor: Laszlo Marki
5. $A$-Solvability and Mixed Abelian Groups, de Ulrich Albrecht, S. Breaz si William Wickless, Comm. in Algebra; editor: K. Rangaswamy
(B) Articole in reviste stiintifice cu vizibilitate internationala
1. Regular submodules of regular Kronecker modules, Csaba Szanto, Mathematica (Cluj), 49(72), no. 1 (2007), 83--89.
2. On some formulas in the Hall algebra of the Kronecker algebra, Csaba Sz\'ant\'o , Acta Mathematica Academiae Paedagogicae Nyiregyhaziensis (AMAPN) (revista EMIS), vol. 23, no. 1 (2007), 1--6.
3. On the completeness of the semihypergroups associated to binary relations, H. Olaru, Cosmin Pelea si I. Purdea, Studia Univ. Babe\c s-Bolyai, Mathematica, vol. LII, no. 2 (2007), 73--85.
4. Products of multialgebras and their fundamental algebras, Cosmin Pelea, acceptata in Proceedings of ICTAMI 2007, Acta Universitatis Apulensis.
5. Direct limits of direct systems of hypergroupoids associated to binary relations, de C. Pelea, si I. Purdea, Advances in Mathematics of Uncertainty, Proc. of the Symposium “Mathematics of Uncertainty”, ECIT 2006, Iasi, Romania, September 20--23, 2006. -
6. Products of hypergroupoids associated to binary relations, de C. Pelea si I. Purdea, Set-valued Mathematics and Applications, 2, 2007, va aparea.
(C) Articole in stadiu de finalizare (preprinturi)
1. Finitistic $n$-costar modules, de S. Breaz.
2. Relatively lifting modules, de Septimiu Crivei
3. Formulas for Kronecker invariants using a representation theoretical approach, Al. Horv\'ath \c si Csaba Szanto
4. Submodules of Kronecker modules via Hall algebras, de A. Hubery si Csaba Szanto
5. On some monic covers and envelopes, de Septimiu Crivei
6. Groups which are determined by subgroup lattices de S. Breaz
Sinteza lucrarilor
Vizite si conferinte
1. Simion Breaz,: Vizita la Dipartimento di Informatica e Comunicazione Università degli Studi dell'Insubria, Seminarul de cercetare coordonat de Prof. Lidia Angeleri Hügel:Seminario di Algebra, "Splitting properties for abelian groups" March 28, 2006 (expunere de 2 ore),
2. Simion Breaz,: Participare la conferinta: Conference on Lattice Theory, In honour of the 70th birthday of George Grätzer and E. Tamás Schmidt AAA 72, Budapest, Hungary, June 6-9, 2006, cu expunerea Every Abelian group is determined by a subgroup lattice
3. Simion Breaz,: International Conference on Abelian Groups and Modules over Commutative Rings Department of Mathematics, University of Connecticut June 11-15, 2007, invited lecture: From tffr groups to mixed groups
4. Simion Breaz,: Participare la 6-th Congress of Romanian Mathematicians, cu expunerea From torsion-free groups to mixed groups.
5. Simion Breaz,: Participare la conferinta ICTAMI 2007 cu expunerea: Every Abelian group is determined by a subgroup lattice II
6. Septimiu Crivei, 1. Vizita la Dipartimento di Informatica e Comunicazione Università degli Studi dell'Insubria, Seminarul de cercetare coordonat de Prof. Lidia Angeleri Hügel, Seminario di Algebra,: "Generalizations of extending modules" March 21, 2006 (expunere de 2 ore)
7. Septimiu Crivei, Conferinta Non-commutative Algebra, Granada, Spain, August 31 -- September 6, 2006, titlu: Extending modules versus purely extending modules
8. Septimiu Crivei, Conferinta Noncommutative Rings and Geometry, Almeria, Spain, September 18--22, 2007, titlu: Extending and lifting type modules
9. Szanto Csaba: Vizita de cercetare la Universitatea din Paderborn (Germania), la invitatia lui Prof. Dr. Henning Krause (invited lecturer pentru 1 luna in mai 2006). Seminarul de cercetare Paderborn Representation Theory Seminar, coordonat de Prof. Henning Krause. Expunerea: Submodules of Kronecker modules via Hall algebras (2 ore, la 18.05.2006).
10. Szanto Csaba: Vizita de cercetare la Universitatea din Bielefeld (Germania) la invitatia lui Prof. Dr. Claus Michael Ringel. Seminarul de cercetare Seminar Darstellungstheorie coordonat de Prof. Claus Michael Ringel: Expunerea: Csaba Szanto Submodules of Kronecker modules via Hall algebras (2 ore, la 19.05.2006).
11. Szanto Csaba: Vizita de cercetare la Universitatea din Koln (Germania) la invitatia lui Prof. Dr. Steffen Konig. Seminarul de cercetare Research Seminar of the Research Group Algebra. Expunerea: Submodules of Kronecker modules via Hall algebras (2 ore, la 23.05.2006).
12. Szanto Csaba: Vizita de cercetare la Universitatea Limburg (Hasselt, Belgia) la invitatia lui Prof. Dr. Michel Van den Bergh in septembrie 2006.
13. Cosmin Pelea Conferinta ECIT2006: Direct limits of direct systems of hypergroupoids associated to binary relations
14. Cosmin Pelea 6-th Congress of Romanian Mathematicians, expunerea: Multialgebras, universal algebras and identities
15. Cosmin Pelea ICTAMI 2007: Products of multialgebras and their fundamental algebras
16. Camelia Dicu Participare la Advanced School and Conference on Representation Theory and Related Topics, 9-27 Ianuarie 2006, Trieste (Italy)
17. Camelia Dicu Participare la Workshop on Triangulated Categories, Leeds 2006, 13-19 august 2006, Leeds (Anglia)
18. Camelia Dicu Participare la scoala de vara Summer School: Microlocal and Geometric Methods in Representation Theory, 17-28 iulie 2006, Guenzburg (Germania)
19. Camelia Dicu Participare la conferinta Some Trends in Algebra 2007, 3-7 septembrie 2007, Praga
20. Camelia Dicu Vizita la institutul Renyi, Budapesta, la Prof. Istvan Agoston
Contact

e-mail:
    bodo AT math.ubbcluj.ro
    ssbreaz AT gmail.com
fax:
    +40 264 691906
mail:
    Simion Breaz
    Babes-Bolyai University
    Faculty of Mathematics and Computer Science
    Str. Mihail Kogalniceanu nr. 1
    RO-400084 Cluj-Napoca
    Romania