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arXiv · 2606.17243

A note on the image of graded multilinear polynomials on upper triangular matrices

Adison Timótio Silva·Felipe Yukihide Yasumura
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Abstract

We investigate the image of polynomials multilinear in graded variables evaluated on the algebra of upper triangular matrices endowed with a group grading. We show that, in general, such an image need not be a vector subspace. However, under the additional assumption that the identity component of the grading is commutative, we prove that the image is always a vector subspace. We further investigate the image of polynomials evaluated on inverse and direct limits of algebras. As a consequence, we prove that the image of a polynomial evaluated on a direct limit of upper triangular matrix algebras whose identity component is commutative is always a vector subspace.

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Adison Timótio Silva, Felipe Yukihide Yasumura. 2026-06-15. A note on the image of graded multilinear polynomials on upper triangular matrices. https://arxiv.org/abs/2606.17243

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