English Version

CONTRACT IDEI 1005 nr. 434/1.10.2007

Descriere succinta a proiectului de cercetare

Proiectul abordeaza mai multe directii actuale de cercetare din algebra necomutativa: algebre Hopf si algebre quasi-Hopf, coinele, categorii braided monoidale, module ingemanate, functori involutivi, arbori binari. Obiectivele proiectului sunt:
1. Obtinerea unor rezultate de clasificare pentru algebre Hopf si algebre quasi-Hopf finit dimensionale.
2. Studiul functorilor tare involutivi pe categorii abstracte si obtinerea unor teoreme generale de tip Moore-Penrose.
3. O teorie a coinelelor in categorii braided monoidale ce admit conuclee, cu aplicatii in teoria structurilor ingemanate.
4. Realizarea unui studiu sistematic al modulelor ingemanate in categorii braided monoidale si descrierea acestora ca si categorii de comodule peste un coinel sau ca si categorii de module peste o algebra monoidala.
5. Unificarea rezultatelor obtinute in ceea ce priveste categoriile de module Hopf peste diverse algebre Hopf sau generalizari ale acestora folosind limbajul si tehnicile categoriilor braided monoidale.
6. Folosind limbajul categoriilor braided monoidale, sa gasim conditii necesare si suficiente pentru care structura de algebra data de un produs smash indus de un braiding local si cea de coalgebra data de un coprodus smash (indusa de asemenea de un braiding local) sa ofere unui anumit obiect o structura de bialgebra si, respective, de algebra Hopf in categorie.
7. O K-teorie pentru coalgebre si algebre Hopf.
8. Abordarea unor probleme de combinatorica privind arborii binari cu ajutorul unor structuri algebrice cuantice.

Echipa de cercetare este alcatuita din trei cercetatori cu experienta si doi doctoranzi.
Tematica celor doua teze de doctorat se incadreaza in obiectivele acestui proiect.

Director de proiect

Prof.dr. Constantin Nastasescu, Facultatea de Matematica si Informatica a Universitatii din Bucuresti.

Seminarii de lucru

 Data  Titlul seminarului
 9.10.2007  Clasificarea algebrelor Hopf finit dimensionale
 12.11.2007  Studiul functorilor involutivi
 10.12.2007  Conceptului de coinel intr-o categorie monoidala
 20.02.2008  Studiul modulelor ingemanate cu ajutorul C-categoriilor
 21.04.2008  Clasificarea algebrelor Hopf punctuate
 22.05.2008  Deformari ale algebrelor graduate
 25.09.2008  Unificarea categoriilor de module Hopf definite peste diferite tipuri de algebra Hopf generalizate
 12.11.2008  Studiul unor ecuatii neliniare
 19.12.2008  Studiul coactiunilor pe spatii de morfisme
 04.02.2009  Studiul algebrelor Hopf co-Frobenius
 30.03.2009  Algebre Hopf incrucisate in categorii braided
 19.06.2009  Coactiuni de algebre Hopf
 24.09.2009  Algebre Hopf de dimensiune finita

Echipa de cercetare a grantului:



Cercetare sprijinita financiar de acest grant:

1. G. Barad, On a remark of Loday about the associahedron and algebraic K-theory, An.St. Univ. Ovidius Constanta vol.16 (1), 5-18, 2008.
2. M. C. Iovanov, C. Nastasescu, J. B. Torrecillas, The Dickson subcategory splitting conjecture for pseudocompact algebras, J. Algebra 320 , 2144-2155, 2008.
3. G. Barad, Finding Eulerian cycle decompositions and the rotation distance between binary trees, Bull. Math. Soc. Sci. Math. Roumanie, Tome 51(99), no 1, 21-38, 2008.
4. S. Dascalescu, Group gradings on diagonal algebras, Arch. Math. 91, No. 3, 212-217 (2008).
5. M. Beattie, D. Bulacu, Braided Hopf algebras obtained from coquasitriangular Hopf algebras, Commun. Math. Phys. 282, 115-160, 2008.
6. F. I. Castano, N. Chifan, C. Nastasescu, Localization on certain Grothendieck categories, Acta Mathematica Sinica, English Series Vol. 25, No. 3, 379-392, 2009.
7. S. Dascalescu, C. Nastasescu, M. Nastasescu, Strongly involutory functors, Commun. Algebra 37, No. 5, 1677-1689, 2009.
8. L. Daus, C. Nastasescu, F. Van Oystaeyen, V-Categories: Applications to Graded Rings, Commun. Algebra 37, No. 9, 3248 - 3258, 2009.
9. M. Beattie, D. Bulacu, On the Antipode of a Co-Frobenius (Co)Quasitriangular Hopf Algebra, Commun. Algebra 37, No. 9, 2981 - 2993, 2009.
10. D. Bulacu, S. Caenepeel, J. B. Torrecillas, Involutory quasi-Hopf algebras, Algebr. Represent. Theory 12, No. 2-5, 257-285, 2009.
11. G. Barad, A nonlinear equation which unify the quantum Yang-Baxter equation and the Pentagonal equation. A Hopf algebra approach, acceptata spre publicare in Bull. Math. Soc. Sci. Math. Roumanie, 2009.

In iunie 2008 G. Barad a obtinut titlul de doctor in matematica la U.B. cu teza "Transformari locale de structuri : flipuri triangulare, rotatii de arbori binari si obstructii combinatoriale de sortare", coordonator Prof. C. Nastasescu.




Raportari

Algebre Hopf in teoria categoriilor, teoria reprezentarilor si combinatorica
(Hopf algebras in category theory, representation theory and combinatorics)

  

UEFISCDI - IDEI 0635, contract 253/05.10.2011

English version:

Descriere succinta a proiectului de cercetare

In acest proiect ne propunem sa continuam studiul algebrelor Hopf si al unor structuri algebrice strans legate de acestea, pe urmatoarele directii de cercetare:

  1. Coactiuni ale unor clase de algebre Hopf pe anumite algebre;
  2. Structuri braided monoidale versus bimonade monoidale;
  3. Proprietati omologice ale algebrelor de incidenta;
  4. Produse incrucisate;
  5. Coomologie ciclica pentru algebre;
  6. Structuri combinatorice cu aplicatii la algebre si coalgebre.

Director de proiect

Prof.dr. Sorin Dascalescu, Facultatea de Matematica si Informatica a Universitatii din Bucuresti.

Echipa de cercetare a grantului:

  • Prof. Dr. Constantin Nastasescu
  • Prof. Dr. Dragos Stefan
  • CS II Dr. Florin Panaite
  • Conf. Dr. Daniel Bulacu
  • Conf. Dr. Doru Staic
  • Asist. Dr. Miodrag Iovanov
  • Doctorand Madalina Barascu
  • Masterand Laura Nastasescu
  • Masterand Adrian Manea

Cercetare sprijinita financiar de acest grant:

  1. S. Dascalescu, M. Iovanov si C. Nastasescu, Quiver Algebras, Path Coalgebras and co-reflexivity, Pacific Journal of Mathematics 262 (2013), 49-79.
  2. M. Barascu si S. Dascalescu, Good gradings on upper block triangular matrix algebras, Comm. Algebra 41 (2013), 4290-4298.
  3. S. Dascalescu, S. Predut si L. Van Wyk, Jordan isomorphisms of generalized structural matrix rings, Linear and Multilinear Algebra Vol. 61, No. 3, martie 2013, 369-376.
  4. S. Dascalescu, M. C. Iovanov, S. Predut, Frobenius structural matrix algebras, Linear Alg. Appl. 439 (2013), 3166-3172.
  5. D. Joita, C. Nastasescu si L. Nastasescu, Recollement of Grothendieck categories. Applications to schemes, Bull. Math. Soc. Sci. Math. Roum. 56 (104), no. 1, 2013, 109-116.
  6. A. Petrescu-Nita si D. M. Staic, Symmetry group of two special types of carbon nanotori, Acta Crystallographica Section A, Volume 69, Part 4, 2013, 435-439.
  7. M. Barascu, Good Z_p^2xZ_pxZ_p-gradings on matrix algebras, Annals of the University of Bucharest (Mathematical Series), 4(2) (2013), 425-431.
  8. M. Iovanov, Complete path algebras and rationals modules, Bull. Math. Soc. Sci. Math. Roum. 56 (2013), 349-364.
  9. F. Panaite, Equivalent crossed products and cross product bialgebras, Comm. Algebra 42(5)(2014), 1937-1952.
  10. F. Panaite, Iterated crossed products, J. Algebra Appl. 13(7) (2014), 1450036 (14 p.).
  11. S. Dascalescu, C. Nastasescu, L. Nastasescu, Frobenius algebras of corepresentations and group-graded vector spaces, J. Algebra 406 (2014), 226-250.
  12. A. Makhlouf, F. Panaite, Yetter-Drinfeld modules for Hom-bialgebras, J. Math. Phys. 55, 013501 (2014).
  13. S. Dascalescu, M. C. Iovanov, Semiperfect and coreflexive coalgebras, Forum Mathematicum 27(5) (2015), 2587-2607.
  14. A. Makhlouf, F. Panaite, Hom-L-R-smash products, Hom-diagonal crossed products and the Drinfeld double of a Hom-Hopf algebra, J. Algebra 441 (2015), 314-343.
  15. M. Iovanov, Triangular matrix coalgebras and applications, Linear and Multilinear Algebra 63(1) (2015), 46-67.
  16. M. Hughes, D. M. Staic, Xie Xiangdong, Classification of a class of nonrigid Carnot groups, Journal of Lie Theory 25 (2015), 717-732.
  17. D. M. Staic, A. Stancu, Operations on the Secondary Hochschild Cohomology, Homology, Homotopy and Applications 17 (2015), 129-146.
  18. D. Bulacu si B. Torrecillas, On Frobenius and separable algebra extensions in monoidal categories. Applications to wreaths, Journal of Noncommutative Geometry 9(3) (2015), 707-774.
  19. G. Graziani, A. Makhlouf, C. Menini, F. Panaite, BiHom-associative algebras, BiHom-Lie algebras and BiHom-bialgebras, Symmetry, Integrability Geom. Methods Appl. (SIGMA) 11 (2015), 086, 34 pagini.
  20. C. Boboc. S. Dascalescu, L. van Wyk, Jordan isomorphisms of 2-torsionfree triangular rings, Linear Multilinear algebra 64(2) (2016), 290-296.
  21. L. Daus, F. Panaite, A new way to iterate Brzezinski crossed products, Colloq. Math. 142(1) (2016), 51-60.
  22. J. Dello, F. Panaite, F. Van Oystaeyen, Y. Zhang, Structure theorems for bicomodule algebras over quasi-Hopfalgebras, Comm. Algebra 44 (2016), 4609-4636.
  23. A. Makhlouf, F. Panaite, Twisting operators, twisted tensor products and smash products for Hom-associative algebras, Glasgow Math. J. 58(3) (2016), 513-538.
  24. D. Stefan, C. Vay, The cohomology ring of the 12-dimensional Fomin-Kirillov algebra, Advances in Mathematics 291 (2016), 584-620.
  25. D. M. Staic, Secondary Hochschild homology, Algebras and Representation theory 19(1) (2016), 47-56.
  26. A. Manea, D. S tefan, Further properties of Koszul pairs and applications, Symmetry Integrability and Geometry-Methods and Applications 12 (2016), Article Number:092.
  27. Bruce R. Corrigan-Salter, Mihai D. Staic, Higher-order and secondary Hochschild cohomology. C. R. Math. Acad. Sci. Paris 354 (2016), no. 11, 1049-1054.
  28. S. Dascalescu, C. Nastasescu, L. Nstasescu, Symmetric algebras in categories of corepresentations and smash products, J. Algebra 465 (2016), 62-80.
  29. M. Iovanov, Z. Mesyan, M. Reyes, Infinite-dimensional diagonalization and semisimplicity, acceptat spre publicare in Israel Journal of Mathematics.
  30. P. Jara, J. L. Pena,D. Stefan, Koszul pairs. Applications, acceptat spre publicare in Journal of Noncommutative Geometry.
  31. D. Bulacu, S. Caenepeel , B.Torrecillas, Frobenius and separable functors for the category of generalized entwined modules. Applications, trimisa spre publicare la Trans. Amer. Math. Soc.
  32. M. Barascu, Counting good gradings on upper block triangular matrix algebras, in curs de revizuire.
  33. D. Bulacu, B. Torrecillas, Galois theory and cleft extensions for monoidal cowreaths. Applications, trimis spre publicare la Proc. London Math. Soc.
  34. A. Manea, D. Stefan, On Koszulity of finite graded posets, acceptat pentru publicare la Journal of Algebra and Its Applications.
  35. F. Panaite, F. Van Oystaeyen, Twisted algebras and Rota-Baxter type operators, acceptat la J. Algebra Appl.
  36. Jacob Laubacher, Mihai D. Staic, Alin Stancu, Bar Simplicial Modules and Secondary Cyclic (Co)homology, in stadiu avansat de finalizare.

Durata proiectului:

01.01.2012-31.12.2016

Buget:

500.000,00 RON/2012
225.745,43 RON/2013
250.000 RON/2014
169.152 RON/2015
355.102,57 RON/2016

Raportare 2012:

Raportare 2013:

Contractare/Raportare 2014:

Contractare/Raportare 2015:

Contractare/Raportare 2016:

2002 - 2015 -- Facultatea de Matematica si Informatica, Universitatea din Bucuresti
Str. Academiei nr. 14, sector 1, C.P. 010014, Bucuresti, Romania
Tel: (4-021) 314 2863, Fax: (4-021) 315 6990, secretariat fmi.unibuc.ro