Matrix representations of finitely generated Grassmann algebras and some consequences

We prove that the m-generated Grassmann algebra can be embedded into a 2 m−1×2 m−1 matrix algebra over a factor of a commutative polynomial algebra in m indeterminates. Cayley-Hamilton and standard identities for n × n matrices over the m-generated Grassmann algebra are derived from this embedding. Other related embedding results are also presented.