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On multilinear polynomial identities for 2×2 matrices in characteristic $2$

  • Iritan Ferreira dos Santos
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Abstract

Let K be an infinite field of characteristic $2$. It is well known that the Lie algebra gl2(K)=M2(K)(−) of all 2× 2 matrices does not admit a finite basis of polynomial identities. Similarly, the finite basis problem for the associative algebra M2(K) remains open. In this note, we obtain two multilinear identities of degree $5$ for M2(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.

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Cite

BibTeXRIS
BibTeX
@misc{santos2026multilinear,
  title = {{On multilinear polynomial identities for $2\times2$ matrices in characteristic $2$}},
  author = {Iritan Ferreira dos Santos},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.25362},
  eprint = {2609.25362},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.25362},
}
RIS
TY  - UNPB
TI  - On multilinear polynomial identities for 2×2 matrices in characteristic $2$
AU  - Iritan Ferreira dos Santos
PY  - 2026
DA  - 2026/09/21
PB  - arXiv
DO  - 10.48550/arxiv.2609.25362
UR  - https://arxiv.org/abs/2609.25362
AB  - Let K be an infinite field of characteristic $2$. It is well known that the Lie algebra gl_2(K)=M_2(K)^(−) of all 2× 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.
ER  -