Fidelis, Claudemir \(\mathbb{Z}\)-graded identities on the infinite-dimensional Grassmann algebra and arithmetic tools: revisited. (English) Zbl 1497.15031 Turk. J. Math. 46, No. 5, SI-2, 1814-1827 (2022). Summary: The main purpose of this paper is to provide a survey of results concerning the \(\mathbb{Z}\)-gradings on the infinite-dimensional Grassmann algebra \(E\) over a field of characteristic zero. First, we provide graded identities and central polynomials for \(E\) equipped with fine gradings on \(E\) by the semigroup \((\mathbb{Z}^\ast, \times)\). We also describe briefly techniques in order to illustrate some important methods to exhibit graded identities and central polynomials of \(E\) for other abelian groups. In particular, over a field of characteristic zero, so-called 2-induced gradings of full support were considered. In order to obtain these descriptions, we strongly use elementary number theory as a tool, providing an interesting connection between this area and PI-Theory. MSC: 15A75 Exterior algebra, Grassmann algebras 16R10 \(T\)-ideals, identities, varieties of associative rings and algebras Keywords:Grassmann algebra; graded identity; graded central polynomial; full support PDFBibTeX XMLCite \textit{C. Fidelis}, Turk. J. Math. 46, No. 5, 1814--1827 (2022; Zbl 1497.15031) Full Text: DOI References: [1] Bahturin YA, Sehgal SK, Zaicev MV. Group gradings on associative algebras. Journal of Algebra 2001; 241 (2): 677-698. doi: 10.1006/jabr.2000.8643 · Zbl 0988.16033 [2] Brandão Jr. AP, Koshlukov P, Krasilnikov A, da Silva EA. The central polynomials for the Grassmann algebra. Israel Journal of Mathematics 2010; 179 (1): 127-144. doi: 10.1007/s11856-010-0074-1 · Zbl 1207.16021 [3] Brandão Jr. A, Fidelis C, Guimarães A. Z -gradings of full support on the Grassmann algebra. Submitted, see also: arXiv: 2009.01870v1, 2020. Journal of Algebra 2022. doi: 10.1016/j.jalgebra.2022.03.014 · Zbl 1504.16039 [4] Centrone L. Z2 -graded identities of the Grassmann algebra in positive characteristic. Linear Algebra and its Applications 2011; 435 (12): 3297-3313. doi: 10.1016/j.laa.2011.06.008 · Zbl 1230.16019 [5] Centrone L. The G -graded identities of the Grassmann Algebra. Archivum Mathematicum 2016; 52 (3): 141-158. doi: 10.5817/AM2016-3-141 · Zbl 1374.16042 [6] Di Vincenzo OM, Koshlukov P, da Silva VRT. On Zp -Graded identities and cocharacters of the Grassmann algebra. Communications in Algebra (2017); 45 (1): 248-262. doi: 10.1080/00927872.2016.1175456 · Zbl 1393.16016 [7] Di Vincenzo OM, da Silva VRT. On Z2 -graded polynomial identities of the Grassmann algebra. Linear Algebra and its Applications (2009); 431 (1-2): 56-72. doi: 10.1016/j.laa.2009.02.005 · Zbl 1225.16009 [8] Drensky V. Free algebras and PI-algebras. Springer, Singapore, 1999. [9] Fidelis C, Guimarães A. Z -gradings on the Grassmann algebra over infinite fields: graded identities and central polynomials. Submitted. [10] Fidelis C, Guimarães AA, Koshlukov P. A note on Z -grading on the Grassmann algebra and elementary number theory. Submitted, see also: arXiv preprint, arXiv: arXiv:2110.06377, 2021. [11] Fonseca LFG. 2 -graded identities of the Grassmann algebra over a finite field. International Journal of Algebra and Computation 2018; 28 (2): 291-307. doi: 10.1142/S0218196718500133 · Zbl 1417.16028 [12] Giambruno A, Zaicev M. Polynomial identities and asymptotic methods. AMS Mathematical Surveys and Mono-graphs Vol. 122, Providence, R.I., 2005. · Zbl 1105.16001 [13] Giambruno A, Koshlukov P. On the identities of the Grassmann algebras in characteristic p > 0 . Israel Journal of Mathematics 2001; 122 (1): 305-316. doi: 10.1007/BF02809905 · Zbl 0990.16022 [14] Guimarães AA. On the support of (R+) -gradings on the Grassmann algebra. Communications in Algebra 2021; 49 (2): 747-762, doi: 10.1080/00927872.2020.1817471 · Zbl 1461.15031 [15] Guimarães AA, Fidelis C, Koshlukov P. Z2 and Z -graded central polynomials of the Grassmann algebra. Interna-tional Journal of Algebra and Computation 2020; 30 (5): 1035-1056. doi: 10.1142/S0218196720500290 · Zbl 1467.15023 [16] Guimarães A, Fidelis C, Dias L. Zq -graded identities and central polynomials of the Grassmann algebra. Linear Algebra and its Applications 2021; 609: 12-36. doi: 10.1016/j.laa.2020.08.014 · Zbl 1470.16046 [17] Guimarães AA, Koshlukov P. Automorphisms and superalgebra structures on the Grassmann algebra. Submitted, see also: arXiv preprint, arXiv :2009 .00175v1, 2020. [18] Guimarães AA, Koshlukov P. Z -graded polynomial identities of the Grassmann algebra. Linear Algebra and its Applications 2021; 617: 190-214. doi: 10.1016/j.laa.2021.02.001. · Zbl 1473.16018 [19] Kemer AR. Varieties and Z2 -graded algebras. Mathematics of the USSR-Izvestiya 1985; 25 (2): 359-374. doi: 10.1070/IM1985v025n02ABEH001285 · Zbl 0586.16010 [20] Kemer AR. Finite basis property of identities of associative algebras. Algebra and Logic 1987; 26: 362-397. · Zbl 0664.16017 [21] Kemer AR. Ideals of Identities of Associative Algebras. AMS Translations of Mathematical Monographs, Vol.87, American Mathematical Society, Providence, RI, 1991. · Zbl 0736.16013 [22] Krakowski D, Regev A. The polynomial identities of the Grassmann algebra. Transactions of the American Math-ematical Society 1973; 181: 429-438. doi: 10.2307/1996643 · Zbl 0289.16015 [23] Latyshev VN. On the choice of basis in a T -ideal. Sibirskii Matematicheskii Zhurnal 1963; 4 (5): 1122-1126 (in Russian). · Zbl 0199.07501 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. 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