On the characteristic polynomial of supermatrices
type: Item
instance of: scholarly article
title: On the characteristic polynomial of supermatrices
zbMATH Open document ID: 0918.16020
DOI: 10.1007/BF02764010
published in: Israel Journal of Mathematics
publication date: 1999-06-14T00:00:00Z
review text: 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 \(G_0\).
Mathematics Subject Classification ID: 16R10
Mathematics Subject Classification ID: 15A75
Mathematics Subject Classification ID: 16R40
Mathematics Subject Classification ID: 16U99
Mathematics Subject Classification ID: 17A70
Mathematics Subject Classification ID: 16S50
zbMATH DE Number: 1239665
zbMATH Keywords: Grassmann algebras
zbMATH Keywords: matrix superalgebras
zbMATH Keywords: integrality
zbMATH Keywords: Cayley-Hamilton theorem
zbMATH Keywords: supermatrices
author: Jenő Szigeti
reviewed by: Matyas Domokos
MaRDI profile type: Publication
cites work: Not all graded identities for Mn(E) come from Mn,n
cites work: Trace Identities and Z/2Z-Graded Invariants
cites work: Supertraces and matrices over Grassmann algebras
cites work: Cayley-Hamilton theorem for \(2\times 2\) matrices over the Grassmann algebra
cites work: On a concept of determinant in the supercase
cites work: VARIETIES AND $ Z_2$-GRADED ALGEBRAS
cites work: On the identities of the matrices over the Grassmann algebra
cites work: Q3735891
cites work: New determinants and the Cayley-Hamilton Theorem for matrices over Lie nilpotent rings
full work available at URL: https://doi.org/10.1007/bf02764010
OpenAlex ID: W2037204187
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description: scientific article; zbMATH DE number 1239665