Graded Identities of Several Tensor Products of the Grassmann Algebra
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Graded Identities of Several Tensor Products of the Grassmann Algebra Lucio Centrone1,2 · Viviane Ribeiro Tomaz da Silva3 Received: 14 May 2020 / Accepted: 16 September 2020 / © Springer Nature B.V. 2020
Abstract Let F be an infinite field of characteristic different from two and E be the unitary Grassmann algebra of an infinite dimensional F -vector space L. Denote by Egr an arbitrary Z2 -grading on E such that the subspace L is homogeneous. We consider Egr ⊗ E ⊗n as a (Z2 × Zn2 )-graded algebra, where the grading on E is supposed to be the canonical one, and we find its graded ideal of identities. Keywords Graded polynomial identities · Grassmann algebra · Tensor product Mathematics Subject Classification (2010) 16R10 · 16R40
1 Introduction The way approaching the study in PI theory changed radically in 1972 after the paper by Regev [18] about the existence of identities in the tensor product of two PI-algebras. Two Presented by: Michel Brion Partially supported by Fundac¸a˜ o de Amparo a` Pesquisa do Estado de S˜ao Paulo (FAPESP) - grants 2015/08961-5 and 2018/02108-7 and Conselho Nacional de Desenvolvimento Cient´ıfico e Tecnol´ogico (CNPq) - grant 308800/2018-4 Partially supported by Conselho Nacional de Desenvolvimento Cient´ıfico e Tecnol´ogico (CNPq) - grant 306534/2016-9 and Fundac¸a˜ o de Amparo a` Pesquisa do Estado de Minas Gerais (FAPEMIG) - grant APQ-01149-18. Viviane Ribeiro Tomaz da Silva
[email protected] Lucio Centrone [email protected]; [email protected] 1
Dipartimento di Matematica, Universit`a degli Studi di Bari “Aldo Moro”, via E. Orabona 4, 70125, Bari, Italy
2
IMECC, Universidade Estadual de Campinas, Rua Sergio Buarque de Holanda 651, 13083-859, Campinas, SP, Brazil
3
Departamento de Matem´atica, Instituto de Ciˆencias Exatas, Universidade Federal de Minas Gerais, Avenida Antˆonio Carlos, 6627, 31270-901, Belo Horizonte, MG, Brazil
L. Centrone, V.R.T. da Silva
decades later Kemer in his famous work [15] proved that the ideal of identities of a given associative PI-algebra over a field of characteristic 0 is the same as the ideal of identities (is PI-equivalent to ∼) of the Grassmann envelope of a finite dimensional superalgebra. We recall if A = A0 ⊕ A1 is a superalgebra, then its Grassmann envelope is G(A) = (A0 ⊗ E 0 ) ⊕ (A1 ⊗ E 1 ), where E = E 0 ⊕ E 1 is the infinite dimensional Grassmann algebra endowed with its canonical Z2 -grading. In particular as a consequence of Kemer’s theory (see [14]) we have the following set of PI-equivalences also called Tensor Product Theorem (TPT) over a field of characteristic 0: E ⊗ E ∼P I M1,1 (E), Ma,b (E) ⊗ E ∼P I Ma+b (E), Ma,b (E) ⊗ Mc,d (E) ∼P I Mac+bd,ad+bc (E), where the algebra Ma,b (E) is a block-subalgebra of Ma+b (E) in which the upper left and lower right blocks are respectively of size a × a and b × b and fulfilled with elements of E 0 and all the other entries are from E 1 . We want to point out that the algebras Mn (F ), Mn (E) and Ma,b (E) are the building blocks of Kemer’s Theory because they generate the only T
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