Matrix representations of finitely generated Grassmann algebras and some consequences

TL;DR

This paper demonstrates that finitely generated Grassmann algebras can be embedded into matrix algebras, enabling derivation of identities and revealing structural properties relevant to algebraic theory.

Contribution

It introduces a new embedding of finitely generated Grassmann algebras into matrix algebras, facilitating the derivation of Cayley-Hamilton and standard identities.

Findings

Grassmann algebra embeds into 2^{m-1}x2^{m-1} matrix algebra

Cayley-Hamilton identities are derived for these embeddings

Additional embedding results are presented

Abstract

We prove that the m-generated Grassmann algebra can be embedded into a 2^{m-1}x2^{m-1} matrix algebra over a factor of a commutative polynomial algebra in m indeterminates. Cayley-Hamilton and standard identities for nxn matrices over the m-generated Grassmann algebra are derived from this embedding. Other related embedding results are also presented.