\(\mathbb Z_2\)-graded identities of the Grassmann algebra in positive characteristic.
The Grassmann (or exterior) algebra plays a prominent role in the theory of algebras with polynomial identities. If the base field \(F\) is of characteristic 0, all T-prime varieties of algebras are described in terms of the Grassmann algebra and of matrix algebras. The polynomial identities of the finite and infinite dimensional Grassmann algebras are well known. Suppose \(V\) is a vector space over \(F\) with a basis \(e_1,e_2,\dots\). One forms the Grassmann algebra \(E=E(V)\) as the vector space with a basis all monomials \(e_{i_1}\cdots e_{i_k}\) where \(i_1<\cdots<i_k\), \(k\geq 0\), and defines a multiplication on \(E\) by \(e_ie_j=-e_je_i\) for all \(i\) and \(j\). The algebra \(E\) has a natural 2-grading (that is by the cyclic group of order 2), \(E=E_0\oplus E_1\) where \(E_0\) is the span of all monomials with even \(k\) while \(E_1\) is the span of the ones with odd \(k\). Thus \(E_0\) is the centre of \(E\); \(E_1\) is the ``anticommutative'' part of \(E\). One may look for other 2-gradings on \(E\). It might well sound intriguing but a complete description of those is not known. On the other hand the situation with such gradings is well understood whenever the vector space \(V\) is homogeneous in the grading. Thus \textit{N. Anisimov} [Commun. Algebra 29, No. 9, 4211-4230 (2001; Zbl 0999.16021)] described the codimensions of the graded identities of \(E\). Later on \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)] studied and described the 2-graded identities of \(E\) assuming \(V\) is homogeneous. Both papers dealt with the characteristic 0 case. The paper under review extends the results above to an arbitrary infinite field of characteristic different from 2. Since \(V\) is homogeneous one may choose a homogeneous basis of \(V\). Thus one considers three types of gradings. The first of them is denoted by \(k*\), and in it the vectors \(e_1,\dots,e_k\) are of degree 1 while all remaining \(e_i\) are of degree 0, The second grading is denoted by \(\infty\); in it \(e_1,e_3,\dots\) are of degree 1 and \(e_2,e_4,\dots\) are of degree 0. The third is denoted by \(k\), it is the antipode of \(k*\). The author describes the graded identities in each of these cases. It turns out that the most difficult to handle is the grading \(k\). In order to describe the identities one passes first to proper (or commutator) polynomials. Here these are considered in their ``graded'' meaning: the variables of degree 0 must appear in commutators only while the odd variables may be inside or outside commutators.
scientific article; zbMATH DE number 5949277