The \(G\)-graded identities of the Grassmann algebra. (Q2828842)

Let \(V\) be an infinite dimensional vector space over a field \(F\) of characteristic 0 and let \(E=E(V)\) be the unitary Grassmann algebra over \(V\). If a finite abelian group \(G\) with identity element \(1_G\) acts on \(V\), then this action is extended diagonally on \(E\) and \(E\) becomes a \(G\)-graded algebra. In the paper under review, the author studies \(G\)-graded polynomial identities and their numerical invariants. The first result of the paper shows that for the description of the ideal of the \(G\)-graded identities of \(E\) it is sufficient to study two cases: If \(| G|\) is odd, then one can reduce the considerations to \(G'\)-gradings, where \(G'\) is a subgroup of \(G\) such that the homogeneous component \(V^{1_{G'}}\) is infinite dimensional and all other components \(V^{g'}\), \(1_{G'}\not=g'\in G'\), are finite dimensional. If \(| G|\) is even, then it is sufficient to consider \(G\)-gradings such that \(V^{g}\) is infinite dimensional if \(g\in G\) satisfies \(g^2=1_G\) and \(V^{g}\) is finite dimensional otherwise. Then the author computes the \(G\)-graded cocharacters and codimensions of \(E\) in the case \(\dim L^{1_G} =\infty\) and \(\dim L^g < \infty\) if \(g\not=1_G\). Explicit results (including the bases of the \(G\)-graded identities) are given when \(G={\mathbb Z}_4\) and \(G={\mathbb Z}_2\times{\mathbb Z}_2\).