On PI-algebras with additional structures: rationality of Hilbert series and Specht's problem
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 with superinvolution. Note that any group algebra is a Hopf algebra and the \(G\)-grading can be viewed as a Hopf algebra coaction. For Specht and local representability problems see [\textit{Y. Karasik}, J. Algebra 457, 194--227 (2016; Zbl 1348.16018)]. The paper is well written and I recommend it to reader also for understanding classical problems.
scientific article; zbMATH DE number 7441517
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