Magnum P.I
One of the approaches for the investigation of the polynomial identities for an algebra over a field \(F\) of characteristic 0 is based on the representation theory of the symmetric and general linear groups. For every \(T\)-ideal, the Young diagrams related to the irreducible characters of the sequence of cocharacters are disposed into a strip. This gives the possibility to exploit the representation theory of Lie superalgebras. Recently, this idea has been used by \textit{A. Berele} and \textit{A. Regev} [J. Algebra 82, 559--567 (1983; Zbl 0517.16013)]. Let \(F\langle X,Y\rangle \) be the free \(\mathbb Z_ 2\)-graded algebra, where the elements of \(X\) and \(Y\) have degree 0 and 1, respectively. The polynomial \(f(x_ 1,...,x_ k,y_ 1,...,y_ m)\in F\langle X,Y\rangle \) is a graded identity for the \(\mathbb Z_ 2\)-graded algebra \(B=B_ 0+B_ 1\) if \(f(a_ 1,...,a_ k,b_ 1,...,b_ m)=0\) for every \(a_ i\in B_ 0\), \(b_ j\in B_ 1\). In the paper under review the author introduces the notion of a magnum of a PI-algebra \(A\). This is the factor-algebra \(F\langle X,Y\rangle /I\), where \(I\) is the ideal of all graded identities of \(A\otimes E\) and \(E\) is the Grassmann algebra with the natural \(\mathbb Z_ 2\)-grading. It turns out that the polynomial identities of every PI-algebra can be regenerated by its magnum. As a consequence, the author obtains the result of Kemer that any PI-algebra \(A\) satisfies all the identities of \(M_ m(E)\) for \(m\) large enough.
representation theory of Lie superalgebras
scientific article; zbMATH DE number 3929208