Growth of algebras with polynomial identities

• Grant number: 15/08961-5
• Support Opportunities: Regular Research Grants
• Start date: September 01, 2015
• End date: November 30, 2017
• Field of knowledge: Physical Sciences and Mathematics - Mathematics - Algebra
• Principal Investigator: Lucio Centrone
• Grantee: Lucio Centrone
• Host Institution: Instituto de Matemática, Estatística e Computação Científica (IMECC). Universidade Estadual de Campinas (UNICAMP). Campinas , SP, Brazil

Abstract

One among the most interesting problems in the area of algebra is the Spechtproblem. In particular, in the case of PI-algebras, the Specht problem has thefollowing form: let A be a PI-algebra over F such that it is finitely generated, thenis it true that the T-ideal of A has a finite number of generators as a T-ideal? In afamous work, Kemer solved into affirmative the previous question in the case F isa eld of characteristic 0. Now we may ask if there is a general method in order toobtain the generators of the T-ideal of any PI-algebra. Then the answer is merelyfar to be provided. In fact we just have a few list of PI-algebras with a well knownset of generators of their T-ideals. That is why we are going to look for other paths.After a result of Regev, it seems more useful to study the Sn-module of multilinearpolynomials that are not polynomial identities for the algebra A, to say, Vn(A).The best path to study the Sn-module Vn(A) is to study the characters of Vn(A)or the cocharacters of A. A similar thing can be said toward the homogeneouspolynomials that are not polynomial identities for A and will be a good idea tostudy the growth of the latter vector space. The responsible researcher alreadyworked and published papers on this topics. He also started collaborations withhigh level researchers such as Vesselin Drensky (BAS-Bulgary), Onofrio Mario DiVincenzo (Università della Basilicata-Italy) and Eli Aljadeff (Technion-Israel). Thegoal is to obtain a general algorithm in order to compute the cocharacters of oneof the most important algebras in the theory, the upper triangular matrices withentries from the infinite dimensional Grassmann algebra. In the same way, we want to develop a theory for the graded Gelfand-Kirillov dimension for graded-semisimplePI-algebras. (AU)

Scientific publications

(7)

(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)

CENTRONE, LUCIO; DA SILVA, VIVIANE RIBEIRO TOMAZ. Graded Identities of Several Tensor Products of the Grassmann Algebra. ALGEBRAS AND REPRESENTATION THEORY, v. 24, n. 6, p. 1441-1458, DEC 2021. (18/02108-7, 15/08961-5)

CENTRONE, LUCIO; DE MELLO, THIAGO CASTILHO. On the factorization of T-G-ideals of graded matrix algebras. BEITRAGE ZUR ALGEBRA UND GEOMETRIE-CONTRIBUTIONS TO ALGEBRA AND GEOMETRY, v. 59, n. 3, p. 597-615, SEP 2018. (15/08961-5, 13/06752-4, 12/16838-0)

CENTRONE, LUCIO. The GK dimension of relatively free algebras of PI-algebras. Journal of Pure and Applied Algebra, v. 223, n. 7, p. 2977-2996, JUL 2019. (15/08961-5, 18/02108-7)

CENTRONE, LUCID. THE G-GRADED IDENTITIES OF THE GRASSMANN ALGEBRA. ARCHIVUM MATHEMATICUM, v. 52, n. 3, p. 141-158, 2016. (15/08961-5, 13/06752-4)

CENTRONE, LUCIO; MARTINO, FABRIZIO. A note on cocharacter sequence of Jordan upper triangular matrix algebra. COMMUNICATIONS IN ALGEBRA, v. 45, n. 4, p. 1687-1695, 2017. (13/06752-4, 15/08961-5)

CENTRONE, LUCIO; MARTINO, FABRIZIO; SOUZA, MANUELA DA SILVA. Specht property for some varieties of Jordan algebras of almost polynomial growth. Journal of Algebra, v. 521, p. 137-165, MAR 1 2019. (18/02108-7, 15/08961-5)

CENTRONE, LUCIO; TOMAZ DA SILVA, VIVIANE RIBEIRO. A note on graded polynomial identities for tensor products by the Grassmann algebra in positive characteristic. INTERNATIONAL JOURNAL OF ALGEBRA AND COMPUTATION, v. 26, n. 6, p. 1125-1140, SEP 2016. (15/08961-5, 13/06752-4)