Groups, rings and algebras: interactions and applications

• Grant number: 09/52665-0
• Support Opportunities: Research Projects - Thematic Grants
• Start date: November 01, 2009
• End date: October 31, 2014
• Field of knowledge: Physical Sciences and Mathematics - Mathematics - Algebra
• Principal Investigator: Francisco Cesar Polcino Milies
• Grantee: Francisco Cesar Polcino Milies
• Host Institution: Instituto de Matemática e Estatística (IME). Universidade de São Paulo (USP). São Paulo , SP, Brazil
• Pesquisadores principais: Jairo Zacarias Goncalves ; Mikhailo Dokuchaev
• Associated research grant(s): 13/19544-0 - Partial actions and partial representations, AV.EXT; 12/20425-3 - Twisted partial actions and the Brauer group, AV.BR; 12/00878-3 - Some classes of semi-perfect rings, AV.EXT; + associated grants 12/03948-2 - Visit to the department of mathematics and statistic, USP, 2012, AV.EXT; 11/51585-3 - Sudarshan Kumar Sehgal | University of Notre Dame - Canada, AV.EXT; 11/50046-1 - Edgar George Goodaire | University of Newfoundland - Canada, AV.EXT; 10/16578-3 - Partial actions and partial representations, cohomology and applications, AV.EXT; 09/54763-0 - Nadiya Gubareni | Technical University of Czestochowa - Polonia, AV.EXT; 09/54126-0 - Sudarshan Kumar Sehgal | University of Notre Dame - Canada, AV.EXT - associated grants
• Associated scholarship(s): 12/20138-4 - About Units of Group Rings and Codes, BP.PD; 12/07610-6 - Partial Schur Multiplier, cohomology and related topics, BP.PD; 12/01554-7 - Partial actions and representations, cohomology and globalization, BP.PD

Abstract

We intend to continue the research on the interactions between group theory and non-commutative algebra, and some of its applications. Our group has worked in these directions for quite some time and has already obtained results that are frequently quoted in the literature. Specific subjects to be studied on the period are, among others: the structure of the group of units of a group ring, determining generators of the free complement, in the case of finite abelian groups, and free pairs of generators, in the general case. We shall also study symmetric and skew symmetric elements in a group algebra. The structure of the group of units of a division ring and the existence of free pairs of special type. Study of globalization of partial actions, generalized crossed products, isomorphism of partial group algebras, partial actions and co-actions of Hopf algebras, partial projective representations and co-homology based on partial actions. Study of codes: cyclic, abelian, metacyclic, etc., using techniques of finite group algebras. (AU)

Scientific publications

(27)

(References retrieved automatically from Web of Science and SciELO through information on FAPESP grants and their corresponding numbers as mentioned in the publications by the authors)

DOKUCHAEV, M.; SIMON, J.. ISOMORPHISMS OF PARTIAL GROUP RINGS. COMMUNICATIONS IN ALGEBRA, v. 44, n. 2, p. 680-696, 2016. (09/52665-0)

CHALOM, GLADYS; FERRAZ, RAUL ANTONIO; GUERREIRO, MARINES. Minimal ideals in finite abelian group algebras and coding theory. SAO PAULO JOURNAL OF MATHEMATICAL SCIENCES, v. 10, n. 2, p. 321-340, DEC 2016. (09/52665-0)

FERREIRA, VITOR O.; GONCALVES, JAIRO Z.; SANCHEZ, JAVIER. Free symmetric algebras in division rings generated by enveloping algebras of Lie algebras. INTERNATIONAL JOURNAL OF ALGEBRA AND COMPUTATION, v. 25, n. 6, p. 1075-1106, SEP 2015. (09/52665-0)

FERREIRA, VITOR O.; GONCALVES, JAIRO Z.. Free symmetric and unitary pairs in division rings infinite-dimensional over their centers. Israel Journal of Mathematics, v. 210, n. 1, p. 297-321, SEP 2015. (09/52665-0)

GONCALVES, JAIRO Z.; PASSMAN, DONALD S.. Free groups in normal subgroups of the multiplicative group of a division ring. Journal of Algebra, v. 440, p. 128-144, OCT 15 2015. (09/52665-0)

FERREIRA, VITOR O.; MURAKAMI, LUCIA S. I.. RATIONALITY OF THE HILBERT SERIES OF HOPF-INVARIANTS OF FREE ALGEBRAS. Proceedings of the American Mathematical Society, v. 142, n. 3, p. 821-826, MAR 2014. (10/50347-9, 09/52665-0)

GOODAIRE, EDGAR G.; MILIES, CESAR POLCINO. ORIENTED INVOLUTIONS AND SKEW-SYMMETRIC ELEMENTS IN GROUP RINGS. JOURNAL OF ALGEBRA AND ITS APPLICATIONS, v. 12, n. 1, FEB 2013. (09/52665-0, 11/50046-1)

MILIES, CESAR POLCINO; DE MELO, FERNANDA DINIZ. On Cyclic and Abelian Codes. IEEE TRANSACTIONS ON INFORMATION THEORY, v. 59, n. 11, p. 7314-7319, NOV 2013. (09/52665-0)

CASTILLO GOMEZ, JOHN H.; MILIES, CESAR POLCINO. LIE PROPERTIES OF SYMMETRIC ELEMENTS UNDER ORIENTED INVOLUTIONS. COMMUNICATIONS IN ALGEBRA, v. 40, n. 12, p. 4404-4419, 2012. (09/52665-0)

FERREIRA, VITOR O.; FORNAROLI, ERICA Z.; GONCALVES, JAIRO Z.. Free algebras in division rings with an involution. Journal of Algebra, v. 509, p. 292-306, SEP 1 2018. (09/52665-0)

ABADIE, F.; DOKUCHAEV, M.; EXEL, R.; SIMON, J. J.. MORITA EQUIVALENCE OF PARTIAL GROUP ACTIONS AND GLOBALIZATION. Transactions of the American Mathematical Society, v. 368, n. 7, p. 4957-4992, JUL 2016. (09/52665-0)

BELL, JASON P.; GONCALVES, JAIRO Z.. Free algebras and free groups in Ore extensions and free group algebras in division rings. Journal of Algebra, v. 455, p. 235-250, JUN 1 2016. (09/52665-0)

GONCALVES, JAIRO Z.; LICHTMAN, A. I.. Free subgroups in division rings generated by group rings of soluble groups. INTERNATIONAL JOURNAL OF ALGEBRA AND COMPUTATION, v. 24, n. 8, p. 1127-1140, DEC 2014. (09/52665-0)

GONCALVES, JAIRO Z.. Constructing free groups in a normal subgroup of the multiplicative group of division rings. Journal of Group Theory, v. 18, n. 5, p. 829-843, SEP 2015. (09/52665-0)

GOODAIRE, EDGAR G.; MILIES, CESAR POLCINO. Jordan nilpotency in group rings. Journal of Group Theory, v. 17, n. 4, p. 541-557, JUL 2014. (12/20208-2, 09/52665-0)

GONCALVES, J. Z.; PASSMAN, D. S.. EXPLICIT FREE GROUPS IN DIVISION RINGS. Proceedings of the American Mathematical Society, v. 143, n. 2, p. 459-468, FEB 2015. (09/52665-0)

GIAMBRUNO, ANTONIO; MILIES, CESAR POLCINO; SEHGAL, SUDARSHAN K.. Star-group identities on units of group algebras: The non-torsion case. FORUM MATHEMATICUM, v. 30, n. 1, p. 213-225, JAN 2018. (09/52665-0)

FERREIRA, VITOR O.; GONCALVES, JAIRO Z.; SANCHEZ, JAVIER. Free symmetric group algebras in division rings generated by poly-orderable groups. Journal of Algebra, v. 392, p. 69-84, OCT 15 2013. (09/52665-0, 09/50886-0)

GONCALVES, J. Z.; PASSMAN, D. S.. INVOLUTIONS AND FREE PAIRS OF BASS CYCLIC UNITS IN INTEGRAL GROUP RINGS. JOURNAL OF ALGEBRA AND ITS APPLICATIONS, v. 10, n. 4, p. 711-725, AUG 2011. (09/52665-0)

GOODAIRE, EDGAR G.; MILIES, CESAR POLCINO. ORIENTED GROUP INVOLUTIONS AND ANTICOMMUTATIVITY IN GROUP RINGS. COMMUNICATIONS IN ALGEBRA, v. 42, n. 4, p. 1657-1667, APR 3 2014. (11/50046-1, 09/52665-0)

GONCALVES, J. Z.; SHIRVANI, M.. A SURVEY ON FREE OBJECTS IN DIVISION RINGS AND IN DIVISION RINGS WITH AN INVOLUTION. COMMUNICATIONS IN ALGEBRA, v. 40, n. 5, p. 1704-1723, 2012. (09/52665-0)

FERRAZ, RAUL ANTONIO; GUERREIRO, MARINES; MILIES, CESAR POLCINO. G-Equivalence in Group Algebras and Minimal Abelian Codes. IEEE TRANSACTIONS ON INFORMATION THEORY, v. 60, n. 1, p. 252-260, JAN 2014. (09/52665-0)

GIAMBRUNO, ANTONIO; MILIES, CESAR POLCINO; SEHGAL, SUDARSHAN K.. Star-group identities and groups of units. ARCHIV DER MATHEMATIK, v. 95, n. 6, p. 501-508, DEC 2010. (09/52665-0)

GONCALVES, J. Z.; TENGAN, E.. FREE GROUP ALGEBRAS IN DIVISION RINGS. INTERNATIONAL JOURNAL OF ALGEBRA AND COMPUTATION, v. 22, n. 5, AUG 2012. (09/52665-0)

MILIES, C. POLCINO; BERGEN, J; CATOIU, S; CHIN, W. Star group identities on units of group algebras. GROUPS, RINGS, GROUP RINGS, AND HOPF ALGEBRAS, v. 688, p. 10-pg., 2017-01-01. (09/52665-0)

GONCALVES, JAIRO Z.; BERGEN, J; CATOIU, S; CHIN, W. Free unit groups in group rings and division rings: My collaboration with Don Passman. GROUPS, RINGS, GROUP RINGS, AND HOPF ALGEBRAS, v. 688, p. 10-pg., 2017-01-01. (09/52665-0)

ABADIE, F.; DOKUCHAEV, M.; EXEL, R.; SIMON, J. J.. MORITA EQUIVALENCE OF PARTIAL GROUP ACTIONS AND GLOBALIZATION. TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY, v. 368, n. 7, p. 36-pg., 2016-07-01. (09/52665-0)