Images of Lie Polynomials on simple Lie algebras
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866914579598016512 |
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| author | Kishnani, Harish Singh, Anupam |
| author_facet | Kishnani, Harish Singh, Anupam |
| contents | A Lie polynomial is an element of a free Lie algebra $\mathcal F_k$ on $k$-generators, which defines a Lie map on a given Lie algebra $L$, by substituting $k$-elements of $L$. Similar to word maps on groups and polynomial maps on algebras, one studies here questions analogous to Waring-like problems, the L'vov-Kaplansky conjecture, etc. In this article, we would like to address a problem for Lie algebras parallel to the one Lubotzky solved (Images of word maps in finite simple groups, Glasg. Math. J., 56, no. 2, 465-469, 2014) for finite simple groups. It is easy to verify that the image of a Lie map is (a) closed under automorphism, and (b) contains $0$. In this article, we prove that for a simple Chevalley algebra over a finite field of ``very good'' characteristic, these two properties are enough to classify all possible subsets that can be the image of a Lie polynomial. The next question is to find such Lie polynomials for a given subset satisfying the two properties. Contrary to the results over an algebraically closed field, we find Lie polynomials in the case of Lie algebra $\mathfrak{sl}_2(q)$, for $q$ odd, which give each $\rm{GL}_2(q)$ conjugacy class together with zero as an image. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_19512 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Images of Lie Polynomials on simple Lie algebras Kishnani, Harish Singh, Anupam Rings and Algebras Group Theory 17B01, 17B20, 15B30 A Lie polynomial is an element of a free Lie algebra $\mathcal F_k$ on $k$-generators, which defines a Lie map on a given Lie algebra $L$, by substituting $k$-elements of $L$. Similar to word maps on groups and polynomial maps on algebras, one studies here questions analogous to Waring-like problems, the L'vov-Kaplansky conjecture, etc. In this article, we would like to address a problem for Lie algebras parallel to the one Lubotzky solved (Images of word maps in finite simple groups, Glasg. Math. J., 56, no. 2, 465-469, 2014) for finite simple groups. It is easy to verify that the image of a Lie map is (a) closed under automorphism, and (b) contains $0$. In this article, we prove that for a simple Chevalley algebra over a finite field of ``very good'' characteristic, these two properties are enough to classify all possible subsets that can be the image of a Lie polynomial. The next question is to find such Lie polynomials for a given subset satisfying the two properties. Contrary to the results over an algebraically closed field, we find Lie polynomials in the case of Lie algebra $\mathfrak{sl}_2(q)$, for $q$ odd, which give each $\rm{GL}_2(q)$ conjugacy class together with zero as an image. |
| title | Images of Lie Polynomials on simple Lie algebras |
| topic | Rings and Algebras Group Theory 17B01, 17B20, 15B30 |
| url | https://arxiv.org/abs/2605.19512 |