Graded identities of several tensor products of the Grassmann algebra
The infinite dimensional Grassmann algebra \(E\) is of significant importance in PI theory. It admits a natural \(\mathbb{Z}_2\)-grading which makes \(E\) the most natural example of a supercommutative superalgebra. This super-structure of \(E\) played a decisive role in the theory developed by Kemer which described the ideals of identities in characteristic 0, see [\textit{A. R. Kemer} and \textit{B. Silver} (ed.), Ideals of identities of associative algebras. Transl. from the Russian by C. W. Kohls. Transl. ed. by Ben Silver. Providence, RI: American Mathematical Society (1991; Zbl 0732.16001)]. The Grassmann algebra admits other gradings by \(\mathbb{Z}_2\). These gradings and their graded identities have been object of intensive study. It should be noted that when the underlying vector space of \(E\) is homogeneous in the grading, there is a complete description of the gradings and of their identities, see [\textit{O. M. Di Vincenzo} and \textit{V. Ribeiro Tomaz da Silva}, Linear Algebra Appl. 431, No. 1--2, 56--72 (2009; Zbl 1225.16009)], in characteristic 0, [\textit{L. Centrone}, Linear Algebra Appl. 435, No. 12, 3297--3313 (2011; Zbl 1230.16019)] (when the base field is infinite and of characteristic different from 2), and [\textit{L. F. Gonçalves Fonseca}, Int. J. Algebra Comput. 28, No. 2, 291--307 (2018; Zbl 1417.16028)] (when the base field is finite). There are some partial results if the underlying vector space of \(E\) is not homogeneous, see [\textit{A. de A. Guimarães} and \textit{P. Koshlukov}, J. Pure Appl. Algebra 227, No. 1, Article ID 107166, 20 p. (2023; Zbl 07567655)]. In the paper under review the authors work over an infinite field of characteristic different from 2, and consider the tensor product \(E\otimes E^{\otimes n}\). Here the first factor \(E\) is graded by \(\mathbb{Z}_2\) in such a way that the generating vector space of \(E\) is homogeneous, and the remaining factors are graded by \(\mathbb{Z}_2\) in the canonical way. They describe the graded identities in each of these cases. Moreover they describe the relatively free graded algebras in all cases.
scientific article; zbMATH DE number 7440309