项目来源

巴西圣保罗研究基金(FAPESP)

项目主持人

Plamen Emilov Kochloukov

项目受资助机构

Instituto de Matemática,Estatística e Computação Científica.Universidade Estadual de Campinas.Campinas,SP,Brazil

立项年度

2025

立项时间

未公开

项目编号

24/20672-8

研究期限

未知 / 未知

项目级别

国家级

受资助金额

未知

学科

Physical Sciences and Mathematics

学科代码

未公开

基金类别

Scholarships in Brazil-Doctorate

关键词

PI-álgebras ; Álgebras com identidades polinomiais ;

参与者

Gabriel Guzatti Vilas Boas

参与机构

未公开

项目标书摘要：In this doctoral research project we focus on problems in the area of Algebra:the theory of algebras with polynomial identities.More precisely,we intend to study the so-called tensor polynomial identities.They were introduced and studied by Huber and Procesi in 2021,2022.In 2021,Huber found a tensor identity of degree 4 for the algebra of matrices of order 2;it resembles the standard polynomial.In 2022 Huber and Procesi began developing methods to study tensor identities in matrix algebras of order$n$,and described a series of such identities for$M_n(F)$,where$F$is a field of characteristic 0.They also began the study of minimal tensor identities for$M_n(F)$.We emphasize here that the description of the minimal tensor identities of$M_n(F)$is still open.(Note that the well-known theorem of Amitsur and Levitzki,obtained in 1951,solves this problem for the ordinary identities of$M_n(F)$,over any field$F$,and even over any unital commutative ring$F$.)We begin with studies of tensor identities for the infinite-dimensional Grassmann algebra$E$over the field$F$.Our goal will be to completely describe the tensor identities of$E$.Next we shall study the tensor identities of$UT_n(F)$,the algebra of upper triangular matrices of order$n$.These two algebras play an important role in PI theory,and the various polynomial identities satisfied by them have been and continue to be the subject of extensive studies.In 2005,Regev and Seeman began studying polynomial identities in twisted(or 2-graded)tensor products.They showed that some classes of prime T-algebras are closed by the twisted tensor product,and conjectured that the twisted tensor product of two prime T-algebras is PI equivalent to the prime T-algebra.This conjecture was confirmed,independently,by Freitas and Koshlukov,and by Di Vincenzo and Nardozza,in 2009.In this project we also intend to study tensor polynomial identities,with the twisted tensor product.Our goals in this direction will be the description of the twisted tensor identities of$E\widehat{\otimes}E$,as well as those of$UT_2(F)\widehat{\otimes}E$.The studies that the student has been doing,since Scientific Initiations,and during his master's degree,will be widely used here.We emphasize that Gabriel has already studied several topics in the theory of algebras with polynomial identities,and is already familiar with the fundamentals of the theory.

Application Abstract: In this doctoral research project we focus on problems in the area of Algebra:the theory of algebras with polynomial identities.More precisely,we intend to study the so-called tensor polynomial identities.They were introduced and studied by Huber and Procesi in 2021,2022.In 2021,Huber found a tensor identity of degree 4 for the algebra of matrices of order 2;it resembles the standard polynomial.In 2022 Huber and Procesi began developing methods to study tensor identities in matrix algebras of order$n$,and described a series of such identities for$M_n(F)$,where$F$is a field of characteristic 0.They also began the study of minimal tensor identities for$M_n(F)$.We emphasize here that the description of the minimal tensor identities of$M_n(F)$is still open.(Note that the well-known theorem of Amitsur and Levitzki,obtained in 1951,solves this problem for the ordinary identities of$M_n(F)$,over any field$F$,and even over any unital commutative ring$F$.)We begin with studies of tensor identities for the infinite-dimensional Grassmann algebra$E$over the field$F$.Our goal will be to completely describe the tensor identities of$E$.Next we shall study the tensor identities of$UT_n(F)$,the algebra of upper triangular matrices of order$n$.These two algebras play an important role in PI theory,and the various polynomial identities satisfied by them have been and continue to be the subject of extensive studies.In 2005,Regev and Seeman began studying polynomial identities in twisted(or 2-graded)tensor products.They showed that some classes of prime T-algebras are closed by the twisted tensor product,and conjectured that the twisted tensor product of two prime T-algebras is PI equivalent to the prime T-algebra.This conjecture was confirmed,independently,by Freitas and Koshlukov,and by Di Vincenzo and Nardozza,in 2009.In this project we also intend to study tensor polynomial identities,with the twisted tensor product.Our goals in this direction will be the description of the twisted tensor identities of$E\widehat{\otimes}E$,as well as those of$UT_2(F)\widehat{\otimes}E$.The studies that the student has been doing,since Scientific Initiations,and during his master's degree,will be widely used here.We emphasize that Gabriel has already studied several topics in the theory of algebras with polynomial identities,and is already familiar with the fundamentals of the theory.