Graded identities of several tensor products of the Grassmann algebra. (English) Zbl 1509.16021
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 [A. R. Kemer and 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 [O. M. Di Vincenzo and V. Ribeiro Tomaz da Silva, Linear Algebra Appl. 431, No. 1–2, 56–72 (2009; Zbl 1225.16009)], in characteristic 0, [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 [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 [A. de A. Guimarães and P. Koshlukov, J. Pure Appl. Algebra 227, No. 1, Article ID 107166, 20 p. (2023; Zbl 1528.16037)].
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.
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.
Reviewer: Plamen Koshlukov (Campinas)
MSC:
| 16R10 | \(T\)-ideals, identities, varieties of associative rings and algebras |
| 16R40 | Identities other than those of matrices over commutative rings |
References:
| [1] | Alves, SM; Koshlukov, P., Polynomial identities of algebras in positive characteristic, J. Algebra, 305, 2, 1149-1165 (2006) · Zbl 1113.16030 · doi:10.1016/j.jalgebra.2006.04.009 |
| [2] | Anisimov, N., \( \mathbb{Z}_p\) ℤp-codimensions of \(\mathbb{Z}_p\) ℤp-identities of Grassmann algebra, Comm. Algebra, 29, 9, 4211-4230 (2001) · Zbl 0999.16021 · doi:10.1081/AGB-100105997 |
| [3] | Azevedo, SS; Fidelis, M.; Koshlukov, P., Tensor product theorems in positive characteristic, J. Algebra, 276, 2, 836-845 (2004) · Zbl 1065.16017 · doi:10.1016/j.jalgebra.2004.01.004 |
| [4] | Azevedo, SS; Fidelis, M.; Koshlukov, P., Graded identities and PI equivalence of algebras in positive characteristic, Commun. Algebra, 33, 4, 1011-1022 (2005) · Zbl 1077.16024 · doi:10.1081/AGB-200053801 |
| [5] | Centrone, L., \( \mathbb{Z}_2\) ℤ2-graded Gelfand-Kirillov dimension of the Grassmann algebra, Internat J. Algebra Comput., 249, 3, 365-374 (2014) · Zbl 1306.16013 · doi:10.1142/S0218196714500167 |
| [6] | Centrone, L., On some recent results about the graded Gelfand-Kirillov dimension of graded PI-algebras, Serdica Math. J., 38, 1-3, 43-68 (2012) · Zbl 1374.16043 |
| [7] | Centrone, L., \( \mathbb{Z}_2\) ℤ2-graded identities of the Grassmann algebra in positive characteristic, Linear Algebra Appl., 435, 12, 3297-3313 (2011) · Zbl 1230.16019 · doi:10.1016/j.laa.2011.06.008 |
| [8] | Centrone, L.; de Mello, TC, On the factorization of TG-ideals of graded matrix algebras, Beitr Algebra Geom., 59, 597-615 (2018) · Zbl 1446.16024 · doi:10.1007/s13366-017-0365-3 |
| [9] | Centrone, L.; da Silva, VRT, A note on tensor product by the Grassmann algebra in positive characteristic, Internat J. Algebra Comput., 26, 6, 1125-1140 (2016) · Zbl 1353.16024 · doi:10.1142/S0218196716500478 |
| [10] | Centrone, L.; da Silva, VRT, On \(\mathbb{Z}_2\) ℤ2-graded identities of UT2(E) and their growth, Linear Algebra Appl., 471, 469-499 (2015) · Zbl 1316.16016 · doi:10.1016/j.laa.2014.12.035 |
| [11] | da Silva e. Silva, DDP, 2-Graded identities for the tensor square of the Grassmann algebra, Linear Multilinear Algebra, 63, 4, 702-712 (2015) · Zbl 1317.16020 · doi:10.1080/03081087.2014.896461 |
| [12] | Di Vincenzo, OM; Nardozza, V., Graded polynomial identities for tensor products by the Grassmann algebra, Comm. Algebra, 31, 3, 1453-1474 (2003) · Zbl 1039.16023 · doi:10.1081/AGB-120017775 |
| [13] | Di Vincenzo, OM; da Silva, VRT, On \(\mathbb{Z}_2\) ℤ2-graded polynomial identities of the Grassmann algebra, Linear Algebra Appl., 431, 1-2, 56-72 (2009) · Zbl 1225.16009 · doi:10.1016/j.laa.2009.02.005 |
| [14] | Kemer, A., Ideals of Identities of Associative Algebras, Trans. Math Monogr, vol. 87 (1991), Providence: Amer. Math. Soc., Providence · Zbl 0736.16013 · doi:10.1090/mmono/087 |
| [15] | Kemer, A., Varieties and \(\mathbb{Z}_2\) ℤ2-graded algebras (Russian), Izv Akad. Nauk SSSR Ser Mat., 48, 5, 1042-1059 (1984) |
| [16] | Popov, AP, Identities of the tensor square of a Grassmann algebra (Russian), Algebra i Logika, 21, 442-471 (1982) · Zbl 0521.16014 |
| [17] | Regev, A., Tensor products of matrix algebras over the Grassmann algebra, J. Algebra, 133, 2, 512-526 (1990) · Zbl 0738.16007 · doi:10.1016/0021-8693(90)90286-W |
| [18] | Regev, A., Existence of identities in A ⊗ B, Israel J. Math., 11, 131-152 (1972) · Zbl 0249.16007 · doi:10.1007/BF02762615 |
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
