On the Z_2-graded codimensions of the Grassmann algebra over a finite field

TL;DR

This paper calculates exact Z_2-graded codimensions for the infinite dimensional Grassmann algebra over a finite field, providing bounds for both homogeneous and non-homogeneous gradings, advancing understanding of algebraic identities.

Contribution

It offers the first exact computation of Z_2-graded codimensions for the Grassmann algebra over finite fields and establishes bounds for non-homogeneous gradings.

Findings

Exact Z_2-graded homogeneous codimensions computed

Bounds established for non-homogeneous codimensions

Results applicable to finite fields with characteristic not 2

Abstract

Let E be the infinite dimensional Grassmann algebra over a finite field F of characteristic not 2. In this paper we deal with the homogeneous Z_2-gradings of E. In particular, we compute an exact value for the Z_2-graded homogeneous codimensions of E, and a lower and an upper bound for the Z_2-graded (non-homogeneous) codimensions of E for each of its Z_2-homogeneous grading.