Graded \(A\)-identities for \(M_{1,1} (E)\). (English) Zbl 1505.15026
Summary: Let \(F\) be a field of characteristic \(0, E\) be the unitary infinite dimensional Grassmann algebra over \(F\) and consider the algebra \(M_{1,1} (E)\) with its natural \(\mathbb{Z}_2\)-grading. We describe the graded \(A\)-identities for \(M_{1,1} (E)\) and we compute its graded \(A\)-codimensions.
MSC:
| 15A75 | Exterior algebra, Grassmann algebras |
| 16W50 | Graded rings and modules (associative rings and algebras) |
References:
| [1] | Drensky, V., Free algebras and PI-algebras (1999), Singapore: Springer, Singapore |
| [2] | Giambruno, A, Zaicev, M. Polynomial identities and asymptotic methods. Vol. 122, Mathematical surveys and monographs. Providence (RI): American Mathematical Society; 2005. · Zbl 1105.16001 |
| [3] | Henke, A.; Regev, A., A-codimensions and A-cocharacters, Israel J Math, 133, 339-355 (2003) · Zbl 1026.16006 |
| [4] | Gonçalves, DJ; Koshlukov, P., A-Identities for the Grassmann algebra: the conjecture of henke and regev, Proc Amer Math Soc, 136, 2711-2717 (2008) · Zbl 1153.16019 |
| [5] | Krakowski, D.; Regev, A., The polynomial identities of the Grassmann algebra, Trans Amer Math Soc, 181, 429-438 (1973) · Zbl 0289.16015 |
| [6] | Giambruno, A.; Koshlukov, P., On the identities of the Grassmann algebras in characteristic p>0, Israel J Math, 122, 305-316 (2001) · Zbl 0990.16022 |
| [7] | Bekh-Ochir, C.; Rankin, SA., The identities and the central polynomials of the infinite dimensional unitary Grassmann algebra over a finite field, Comm Algebra, 39, 3, 819-829 (2011) · Zbl 1226.16016 |
| [8] | Popov, AP., Identities for the tensor square of the Grassmann algebra, Algebra and Logic, 21, 442-471 (1982) · Zbl 0521.16014 |
| [9] | Regev, A., Tensor products of matrix algebras over the Grassmann algebra, J Algebra, 133, 2, 512-526 (1990) · Zbl 0738.16007 |
| [10] | Naves, FA; Talpo, HL., Minimun degree of an A-identity of \(####\), Internat J Algebra Comput, 30, 6, 1237-1256 (2020) · Zbl 1459.16025 |
| [11] | Kemer, A., Varieties and \(####\)-graded algebras, Math USSR Izv, 25, 359-374 (1985) · Zbl 0586.16010 |
| [12] | Kemer, A., Finite basis property of identities of associative algebras, Algebra Logic, 26, 362-397 (1987) · Zbl 0664.16017 |
| [13] | Kemer, AR. Ideals of identities of associative algebras. Vol. 87, Translation of mathematical monographs. Providence (RI): American Mathematical Society; 1991. · Zbl 0751.16007 |
| [14] | Di Vincenzo, OM; da Silva, VRT., On \(####\)-graded identities of the Grassmann algebra, Linear Algebra Appl, 431, 1-2, 56-72 (2009) · Zbl 1225.16009 |
| [15] | Centrone, L., \(####\)-graded identities of the Grassmann algebra in positive characteristic, Linear Algebra Appl, 435, 12, 3297-3313 (2011) · Zbl 1230.16019 |
| [16] | Di Vincenzo, OM., On the graded identities of \(####\), Israel J Math, 80, 323-335 (1992) · Zbl 0784.16015 |
| [17] | Koshlukov, P.; de Azevedo, SS., Graded identities for T-prime algebras over fields of positive characteristic, Israel J Math, 128, 157-176 (2002) · Zbl 1015.16023 |
| [18] | Brandão Jr, AP; Gonçalves, DJ; Koshlukov, P., Graded A-identities for the matrix algebra of order two, Int J Algebra Comput, 26, 8, 1617-1631 (2016) · Zbl 1366.16018 |
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.
