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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.

##### 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
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