On sums of \textit{gr}-PI algebras (Q6051141)

\textit{M. Kȩpczyk} [Isr. J. Math. 221, No. 1, 481--487 (2017; Zbl 1396.16015)] proved that if an algebra \(A\) is the vector space sum of two subalgebras \(A=B+C,\) and if \(B\) and \(C\) each satisfy polynomial identities, then \(A\) will also satisfy a polynomial identity. The current paper considers the case in which \(A\) is graded by a group \(G,\) and the subalgebras \(B\) and \(C\) are homogeneous. The two main results are that if \(B\) and \(C\) each satisfy graded identities then \(A\) may not satisfy any graded identity; but, if in addition one or both of them is an ideal of \(A,\) then \(A\) will satisfy a graded identity. The counterexamples in the first of these theorems also extend to Lie algebras. The authors also define and discuss graded semi-identities. These are polynomials in which each variable has not only a \(G\)-degree, but also a restriction as to whether it is to be evaluated on \(B\) or \(C\). Kȩpczyk proved that if an algebra \(A\) is the vector space sum of two subalgebras \(A=B+C,\) and if \(B\) and \(C\) each satisfy polynomial identities, then \(A\) will also satisfy a polynomial identity. The current paper considers the case in which \(A\) is graded by a group \(G,\) and the subalgebras \(B\) and \(C\) are homogeneous. The two main results are that if \(B\) and \(C\) each satisfy graded identities then \(A\) may not satisfy any graded identity; but, if in addition one or both of them is an ideal of \(A,\) then \(A\) will satisfy a graded identity. The counterexamples in the first of these theorems also extend to Lie algebras. The authors also define and discuss graded semi-identities. These are polynomials in which each variable has not only a \(G\)-degree, but also a restriction as to whether it is to be evaluated on \(B\) or \(C\).