Tensor polynomial identities

• Grant number: 24/20672-8
• Support Opportunities: Scholarships in Brazil - Doctorate
• Start date: March 01, 2025
• End date: February 28, 2029
• Field of knowledge: Physical Sciences and Mathematics - Mathematics - Algebra
• Principal Investigator: Plamen Emilov Kochloukov
• Grantee: Gabriel Guzatti Vilas Boas
• Host Institution: Instituto de Matemática, Estatística e Computação Científica (IMECC). Universidade Estadual de Campinas (UNICAMP). Campinas , SP, Brazil

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.