On the minimal varieties of PI-algebras graded by finite cyclic groups

Let F be an algebraically closed field of characteristic zero and G a finite cyclic group. Let be a variety of associative G-graded PI-algebras over F of finite basic rank. In this paper, we prove that if is minimal with respect to a given G-exponent, then there exist finite-dimensional G-simple F-a...

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Bibliographic Details
Published in:Linear & multilinear algebra 2022-12, Vol.70 (20), p.5790-5814
Main Authors: Pinto, Marcos Antônio da Silva, da Silva, Viviane Ribeiro Tomaz
Format: Article
Language:English
Subjects:
finite cyclic group
G-exponent
Graded algebras
graded polynomial identities
Mathematical analysis
Matrix algebra
minimal varieties
PI-algebras
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Summary:Let F be an algebraically closed field of characteristic zero and G a finite cyclic group. Let be a variety of associative G-graded PI-algebras over F of finite basic rank. In this paper, we prove that if is minimal with respect to a given G-exponent, then there exist finite-dimensional G-simple F-algebras such that is generated by a suitable G-graded upper block triangular matrix algebra endowed with an elementary grading and where the diagonal blocks are given by the 's. Moreover, for a fixed m-tuple of finite-dimensional G-simple F-algebras, we prove the converse of the above result for some important classes of G-graded algebras endowed with elementary gradings. In particular, we conclude that the variety generated by A is minimal when A has one or two G-simple blocks as well whenever all (except for at most one) the G-simple components of A are G-regular.
ISSN:0308-1087
1563-5139
DOI:10.1080/03081087.2021.1930990