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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.

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
Full Text: DOI arXiv
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