×

zbMATH — the first resource for mathematics

Matrix representations of finitely generated Grassmann algebras and some consequences. (English) Zbl 1344.16023
The authors prove that the \(m\)-generated Grassmann algebra can be embedded into a \(2^{m-1}\times 2^{m-1}\) matrix algebra over a factor of a commutative polynomial algebra in \(m\) indeterminates. Cayley-Hamilton and standard identities for \(n\times n\) matrices over the \(m\)-generated Grassmann algebra are derived using this embedding.
Theorem. Let \(A\) be an \(n\times n\) matrix with coefficients belonging to the \(m\)-generated Grassmann algebra over a field \(K\) of characteristic zero. Then \(A\) satisfies an identity of the form: \[ A^{2^{m-1}n}+c_1A^{2^{m-1}n-1}+\cdots+c_{2^{m-1}n-1}A+c_{2^{m-1}n}I_n=0,\quad c_i\in K. \] The authors also present embedding results for skew polynomial rings.

MSC:
16R10 \(T\)-ideals, identities, varieties of associative rings and algebras
16R20 Semiprime p.i. rings, rings embeddable in matrices over commutative rings
15A75 Exterior algebra, Grassmann algebras
PDF BibTeX XML Cite
Full Text: DOI arXiv
References:
[1] Domokos, M., Eulerian polynomial identities and algebras satisfying a standard identity, Journal of Algebra, 169, 913-928, (1994) · Zbl 0821.16029
[2] M. Domokos and M. Zubor, Commutative subalgebras of the Grassmann algebra, arXiv:1403.2916v2. · Zbl 1318.15011
[3] A. R. Kemer, Ideals of Identities of Associative Algebras, Translations of Mathematical Monographs, Vol. 87, American Mathematical Society, Providence, RI. 1991. · Zbl 0732.16001
[4] Mirzakhani, M., A simple proof of a theorem of Schur, American Mathematical Monthly, 105, 260-262, (1998) · Zbl 0916.15004
[5] L. H. Rowen, Polynomial Identities in Ring Theory, Pure and Applied Mathematics, Vol. 84, Academic Press, New York, 1980. · Zbl 0461.16001
[6] Sehgal, S.; Szigeti, J., matrices over centrally ℤ_{2}-graded rings, Beiträge zur Algebra und Geometrie, 43, 399-406, (2002) · Zbl 1016.15013
[7] Szigeti, J., New determinants and the Cayley-Hamilton theorem for matrices over Lie nilpotent rings, Proceedings of the American Mathematical Society, 125, 2245-2254, (1997) · Zbl 0888.16011
[8] J. Szigeti, Embedding truncated skew polynomial rings into matrix rings and embedding of a ring into 2 × 2 supermatrices, arXiv:1307.1783.
[9] Szigeti, J.; vanWyk, L., determinants for n×n matrices and the symmetric Newton formula in the 3 × 3 case, Linear and Multilinear Algebra, 62, 1076-1090, (2014) · Zbl 1309.15014
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.