Year: 2026
Type: preprint
Abstract: Let $K$ be an infinite field of characteristic $2$. It is well known that the Lie algebra $\mathfrak{gl}_2(K)=M_2(K)^{(-)}$ of all $2\times 2$ matrices does not admit a finite basis of polynomial identities. Similarly, the finite basis problem for the associative algebra $M_2(K)$ remains open. In this note, we obtain two multilinear identities of degree $5$ for $M_2(K)$ and prove that, together with the standard polynomial of degree $4$, they generate all multilinear identities of degree up to $7$. Moreover, we prove that none of the three generators can be omitted.
Source: arXiv (Cornell University)
Author: Santos, Iritan Ferreira dos
Language: English
Topic: Advanced Topics in Algebra
Subfield: Algebra and Number Theory
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7214083305