Grant CEEX19

Research project
"Echivalences and dualities between module categories, applications and generalizations",
October 2005-September2007.

Financed by National Authority for Scientific Research (ANCS)

Algebra Department, Faculty of Mathematics and Computer Science
"Babes-Bolyai" University, Cluj-Napoca

Team:

Simion Breaz (project manager),
Septimiu Crivei,
Cosmin Pelea,
Csaba Szanto,
Camelia Dicu.

Objectives:

(I) Scientific objectives

1) Characterization of self-small abelian groups with substitution and simplification properties.
2) Establishing dualities between different module categories. The aim is the construction of dualities different from those induced by costar modules (introduced by Colby and Fuller).
3) Description of submodule categories using techniques initiated by Ringel and Arnold for subgroup categories.
4) Characterization of self-small groups (modules) with different splitting properties. The subject of study are the Baer and Szele splitting properties and their characterization with the help of equivalences and dualities.
5) Characterization of equivalences and dualities between functor categories. The aim is the generalization of Morita, Gruson-Jensen, Gabriel-Popescu type theorems for different additive categories, quotient categories of some functor categories.
6) Establishing connections between the existence of specific dualities, submodule categories, generalizations of endoprimal universal algebras and homogeneous functions.

(II) Raising the international impact of the research

1. Publication of the results in journals with high international impact (ISI cited and/or reviewed by Mathematical Reviews or Zentralblatt fur Mathematik). 2. Presentation of the results on international research seminars and conferences.

(III) Improving the research conditions

1. Completion of the research infrastructure by
-- assuring the bibliographical background (books, journals, etc.)
-- assuring the material background of the research (computers, office consumables etc.)
2. Improving the level of knowledge of the team members and other young master and PhD students about the newest research directions

Achievements:

(A) Papers in ISI journals

1. Cancellation Properties for Quotient Divisible Groups, de U. Albrecht (Auburn University), S. Breaz, C. Vinsohaler (University of Connecticut) and 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 and 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 and W. Wickless (University of Connecticut), accepted by Houston Journal of Mathematics, editor: Jutta Hausen.
4. Every Abelian group is determined by a subgroup lattice, S. Breaz and Gr. Calugareanu, Studia Scientiarum Mathematicarum Hungaricae, doi:10.1556/SScMath.2007.1046; editor: Laszlo Marki

5. 4. $A$-Solvability and Mixed Abelian Groups, by Ulrich Albrecht, S. Breaz and William Wickless, Comm. Algebra, editor: K. Rangaswamy

(B) Papers in journals with international impact
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 Szanto , Acta Mathematica Academiae Paedagogicae Nyiregyhaziensis (AMAPN) (an EMIS journal), vol. 23, no. 1 (2007), 1--6.
3. On the completeness of the semihypergroups associated to binary relations, H. Olaru, Cosmin Pelea and 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, accepted in Proceedings of ICTAMI 2007, Acta Universitatis Apulensis.
5. Direct limits of direct systems of hypergroupoids associated to binary relations, C. Pelea, and 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, C. Pelea and I. Purdea, Set-valued Mathematics and Applications, 2, 2007, to appear.

(C) Preprints
1. Finitistic $n$-costar modules, by S. Breaz.
2. Relatively lifting modules, by Septimiu Crivei
3. Formulas for Kronecker invariants using a representation theoretical approach, by Al. Horvath and Csaba Szanto
4. Submodules of Kronecker modules via Hall algebras, by A. Hubery and Csaba Szanto
5. On some monic covers and envelopes, by Septimiu Crivei
6. Groups which are determined by subgroup lattices by S. Breaz

Abstracts

Research visits and conferences. Talks

1. Simion Breaz, visit at Dipartimento di Informatica e Comunicazione Università degli Studi dell'Insubria. Talk presented at the research seminar coordonated by Prof. Lidia Angeleri Hügel, Seminario di Algebra: Splitting properties for abelian groups March 28, 2006 (2 hours talk),
2. Simion Breaz, talk at the 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: 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, talk at the 6-th Congress of Romanian Mathematicians: From torsion-free groups to mixed groups.
5. Simion Breaz, talk at the ICTAMI 2007: Every Abelian group is determined by a subgroup lattice II
6. Septimiu Crivei, visit at Dipartimento di Informatica e Comunicazione Università degli Studi dell'Insubria. Talk presented at the research seminar coordonated by Prof. Lidia Angeleri Hügel, Seminario di Algebra: Generalizations of extending modules March 21, 2006 (2 hours talk)
7. Septimiu Crivei, talk at the Non-commutative Algebra conference, Granada, Spain, August 31 -- September 6, 2006: Extending modules versus purely extending modules
8. Septimiu Crivei, talk at the Noncommutative Rings and Geometry conference, Almeria, Spain, September 18--22, 2007: Extending and lifting type modules
9. Szanto Csaba, visit at Paderborn University, invited by Prof. Dr. Henning Krause (for 1 month, May 2006). Talk presented at the research seminar Paderborn Representation Theory Seminar: Submodules of Kronecker modules via Hall algebras ,May 18, 2006
10. Szanto Csaba, visit at Bielefeld University, invited by Prof. Dr. Claus Michael Ringel. Talk presented at the research seminar Seminar Darstellungstheorie Submodules of Kronecker modules via Hall algebras ,May 19, 2006.
11. Szanto Csaba, visit at Koeln University, invited by Prof. Dr. Steffen Konig. Talk presented at the research seminar Research Seminar of the Research Group Algebra: Submodules of Kronecker modules via Hall algebras , May 23, 2006.
12. Szanto Csaba, visit at Limburg University, invited by Prof. Dr. Michel Van den Bergh in September 2006.
13. Cosmin Pelea, talk at the conference ECIT2006: Direct limits of direct systems of hypergroupoids associated to binary relations
14. Cosmin Pelea, talk at the 6-th Congress of Romanian Mathematicians: Multialgebras, universal algebras and identities
15. Cosmin Pelea, talk at the ICTAMI 2007: Products of multialgebras and their fundamental algebras
16. Camelia Dicu, participation Advanced School and Conference on Representation Theory and Related Topics, 9-27 Ianuarie 2006, Trieste (Italy)
17. Camelia Dicu, participation Workshop on Triangulated Categories, Leeds 2006, 13-19 august 2006, Leeds (Anglia)
18. Camelia Dicu, participation Summer School: Microlocal and Geometric Methods in Representation Theory, 17-28 iulie 2006, Guenzburg (Germania)
19. Camelia Dicu, participation Some Trends in Algebra 2007, 3-7 septembrie 2007, Praga
20. Camelia Dicu, visit at Renyi Institute, Budapest, invited by 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