On PI-algebras with additional structures: rationality of Hilbert series and Specht's problem
type: Item
instance of: scholarly article
title: On PI-algebras with additional structures: rationality of Hilbert series and Specht's problem
zbMATH Open document ID: 1497.16023
author: Lucio Centrone
author: Alejandro Estrada
author: Antonio Ioppolo
published in: Journal of Algebra
publication date: 2021-12-09T00:00:00Z
review text: One of the main topics in PI-theory are Specht-type problems: the classification of \(T\)-prime varieties, finite basis of identities problem (Specht problem), local representability of relatively free algebras and the Procesi question: to prove the rationality of the Hilbert series of the relatively free algebra of a given PI-algebra [\textit{C. Procesi}, Rings with polynomial identities. Marcel Dekker, Inc., New York, NY (1973; Zbl 0262.16018)]. For characteristic zero the Specht problem and local representability problem was positively solved by [\textit{A. R. Kemer}, Transl., Ser. 2, Am. Math. Soc. 148, 65--71 (1991; Zbl 0736.16013)]. He also established the local finite basis property and local representability for infinite fields [Math. USSR, Izv. 37, No. 1, 69--96 (1990; Zbl 0784.16016); translation from Izv. Akad. Nauk SSSR, Ser. Mat. 54, No. 4, 726--753 (1990)]. The Procesi problem has been solved (see [\textit{A. Ya. Belov}, Russ. Math. Surv. 52, No. 2, 394--395 (1997; Zbl 0919.16018); translation from Usp. Mat. Nauk 52, No. 2, 153--154 (1997); \textit{E. Aljadeff} and the reviewer, Bull. Lond. Math. Soc. 44, No. 3, 520--532 (2012; Zbl 1259.16023)]. In positive characteristic the global Speht problem has a negative solution [\textit{A. Ya. Belov}, Fundam. Prikl. Mat. 5, No. 1, 47--66 (1999; Zbl 0964.16024)]. Slightly latter other counterexamples were presented in [\textit{A. Y. Belov}, Izv. Math. 74, No. 1, 1--126 (2010; Zbl 1208.16022); translation from Izv. Ross. Akad. Nauk, Ser. Mat. 74, No. 1, 3--134 (2010)]. This paper is devoted to the Procesi problem in the Hopf algebra co-action case. The authors consider a field \(F\) of characteristic \(0\) and prove the rationality of the Hilbert series of the PI-algebra \(A\) over \(F\) both in the case \(A\) is a superalgebra with superinvolution and when a finite dimensional semisimple Hopf algebra acts on \(A\). Along the way, they give a proof of the Specht's problem in the case where\(A\) is a superalgebra…
Mathematics Subject Classification ID: 16R10
Mathematics Subject Classification ID: 16R50
Mathematics Subject Classification ID: 16W55
Mathematics Subject Classification ID: 16T05
zbMATH DE Number: 7441517
zbMATH Keywords: Hilbert series
zbMATH Keywords: polynomial identities
zbMATH Keywords: superinvolutions
zbMATH Keywords: Hopf algebras, Specht type problems
reviewed by: A. Ya Belov
MaRDI profile type: Publication
full work available at URL: j.jalgebra.2021.11.005
OpenAlex ID: W3212505045
cites work: Multialternating graded polynomials and growth of polynomial identities.
cites work: Polynomial identities with involution, superinvolutions and the Grassmann envelope
cites work: Graded polynomial identities and exponential growth
cites work: Representability and Specht problem for \(G\)-graded algebras.
cites work: Hilbert series of PI relatively free \(G\)-graded algebras are rational functions.
cites work: Non-commutative graded algebras and their Hilbert series
cites work: Q5563439
cites work: Group gradings on superinvolution simple superalgebras.
cites work: On the rationality of Hilbert series of relatively free algebras
cites work: The Gel'fand-Kirillov dimension of relatively free associative algebras
cites work: Monomial algebras
cites work: Polynomial identities on superalgebras and exponential growth
cites work: Generic verbally prime pi-algebras and their gk-dimensions
cites work: Homogeneous polynomial identities
cites work: Asymptotic behaviour of codimensions of p. i. algebras satisfying Capelli identities
cites work: The GK dimension of relatively free algebras of PI-algebras
cites work: Q2963538
cites work: A NOTE ON GRADED GELFAND–KIRILLOV DIMENSION OF GRADED ALGEBRAS
cites work: Q4039887
cites work: Q4520725
cites work: Codimensions of T-ideals and Hilbert series of relatively free algebras
cites work: Q4270218
cites work: Q4700803
cites work: GK-dimension of $2\times 2$ generic Lie matrices
cites work: Varieties of algebras with superinvolution of almost polynomial growth
cites work: Superalgebras with involution or superinvolution and almost polynomial growth of the codimensions
cites work: Graded polynomial identities and codimensions: computing the exponential growth.
cites work: Q5714236
cites work: Exponential codimension growth of PI algebras: an exact estimate
cites work: On the Lie structure of the skew elements of a simple superalgebra with superinvolution
cites work: Lie algebras simple with respect to a Taft algebra action
cites work: The exponent for superalgebras with superinvolution
cites work: Lie superalgebras
cites work: Q4658075
cites work: Rings with a polynomial identity
cites work: Kemer's theory for \(H\)-module algebras with application to the PI exponent
cites work: Q3789721
cites work: Identities of associative algebras
cites work: Q4705362
cites work: Q5840229
cites work: Computing minimal free resolutions of right modules over noncommutative algebras
cites work: Monomial right ideals and the Hilbert series of noncommutative modules
cites work: An Associative Orthogonal Bilinear Form for Hopf Algebras
cites work: STABLE JACOBSON RADICALS AND SEMIPRIME SMASH PRODUCTS
cites work: Noncommutative Noetherian rings. With the cooperation of L. W. Small.
cites work: Biinvertible actions of Hopf algebras
cites work: Primitive superalgebras with superinvolution
cites work: Simple Jordan superalgebras with semisimple even part.
cites work: Hopf Algebras
cites work: Q2716065
cites work: Q3243334
cites work: Identities of PI-Algebras Graded by a Finite Abelian Group
cites work: Q5590089
DOI: 10.1016/J.JALGEBRA.2021.11.005
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