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Polynomial identities and superinvolutions

Grant number: 18/17464-3
Support type:Scholarships in Brazil - Post-Doctorate
Effective date (Start): March 01, 2019
Effective date (End): February 28, 2021
Field of knowledge:Physical Sciences and Mathematics - Mathematics - Algebra
Principal Investigator:Plamen Emilov Kochloukov
Grantee:Antonio Ioppolo
Home Institution: Instituto de Matemática, Estatística e Computação Científica (IMECC). Universidade Estadual de Campinas (UNICAMP). Campinas , SP, Brazil
Associated research grant:14/09310-5 - Algebraic structures and their representations, AP.TEM


The main object of study of this research project is the theory of polynomial identities for associative algebras. We are interested in algebras endowed with a particular linear map called superinvolution.The superinvolutions are a natural generalization of the involutions and they play a prominent role in the setting of Lie and Jordan superalgebras as shown by Kac in around 1977, and later on by Racine and Zelmanocv, in 2003. Among the most important numerical invariants of a T-ideal (that is the ideal of identities of an algebra) is its codimension sequence. A fundamental theorem due to Regev proves that in the case of associative algebra, this sequence grows at most exponentially. Around 1980, Amitsur conjectured that the n-th roots of the n-th codimensions of a PI algebra converge and that the limit is always an integer. This was proved to be true by Giambruno and Zaicev in 1999. Later on this was transferred to other classes of algebras as well. Our first goal will be to study the structure of the ideal of identities of graded algebras with graded involutions. We want to prove that any such algebra satisfies the same identities as the Grassmann envelope of a finite-dimensional graded superalgebra. This will be an analogue of an important theorem of Kemer. Such a result would make it possible to prove that the PI exponent exists and is an integer in the case of algebras with graded involution. (The result holds for finitely genearted algebras as of now.)The simple superalgebras with superinvolution were classified by Racine. We aim at studying the superalgebra M(2,1) equipped with the orthosymplectic superinvolution, and determine the corresponding polynomial identities. The last goal of the project will be to study associative algebras with trace and their trace identities. Trace identities for the matrix algebras were studied and described in the seminal works of Preocesi and Razmyslov. We aim at classifying the algebras with trace whose trace codimensions (pure and mixed) are of polynomial growth. We also aim at describing the varieties of algebras with trace of almost polynomial growth (that is the varieties whose growth is exponential but easch proper subvariety is of polynomial growth). The méthods will be based on structure theory of algebras as well as on then representation theory of the symmetric and general linear group, and also on the theory of Hamilton-Cayley algebras. Os resultados da pesquisa serão publicados em revistas especializadas.

Scientific publications
(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)
IOPPOLO, ANTONIO. Superalgebras with superinvolution or graded involution with colengths sequence bounded by 3. INTERNATIONAL JOURNAL OF ALGEBRA AND COMPUTATION, v. 30, n. 4, p. 821-838, JUN 2020. Web of Science Citations: 0.

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