On the characteristic polynomial of supermatrices
Let \(G=G_0+G_1\) be the infinite dimensional Grassmann algebra over a field of characteristic zero with its usual \(\mathbb{Z}_2\)-grading, and \(M_{n,k}(G)\) the \(n\times n\) matrix superalgebra with entries from \(G_0\) in the left upper \(k\times k\) and the right lower \((n-k)\times(n-k)\) blocks, and entries from \(G_1\) in all other positions (\(1\leq k<n\)). The main result is that \(M_{n,k}(G)\) is integral of degree \(n^2\) over its center \(G_0\). The proof is based on the Cayley-Hamilton theorem for Lie nilpotent rings invented by the same author [in Proc. Am. Math. Soc. 125, No. 8, 2245-2254 (1997; Zbl 0888.16011)], where he showed that \(M_n(G)\) is integral of degree \(2n^2\) over its center. The degree of integrality of supermatices is improved by showing that the coefficients of the characteristic polynomials of supermatrices lie in \(G_0\).
scientific article; zbMATH DE number 1239665
scientific article