Automorphisms and derivations of a universal left-symmetric enveloping algebra
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866915365604294656 |
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| author | Zhangazinova, D. Naurazbekova, A. Umirbaev, U. |
| author_facet | Zhangazinova, D. Naurazbekova, A. Umirbaev, U. |
| contents | Let $A_n$ be an $n$-dimensional algebra with zero multiplication over a field $K$ of characteristic $0$. Then its universal (multiplicative) enveloping algebra $U_n$ in the variety of left-symmetric algebras is a homogeneous quadratic algebra generated by $2n$ elements $l_1,\ldots,l_n,r_1,\ldots,r_n$, which contains both the polynomial algebra $L_n=K[l_1,\ldots,l_n]$ and the free associative algebra $R_n=K\langle r_1,\ldots,r_n\rangle$. We show that the automorphism groups of the polynomial algebra $L_n$ and the algebra $U_n$ are isomorphic for all $n\geq 2$, based on a detailed analysis of locally nilpotent derivations. In contrast, we show that this isomorphism does not hold for $n=1$, and we provide a complete description of all automorphisms and locally nilpotent derivations of $U_1$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2506_24051 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Automorphisms and derivations of a universal left-symmetric enveloping algebra Zhangazinova, D. Naurazbekova, A. Umirbaev, U. Rings and Algebras 13F20, 14R10, 16D25, 16W20, 17D25 Let $A_n$ be an $n$-dimensional algebra with zero multiplication over a field $K$ of characteristic $0$. Then its universal (multiplicative) enveloping algebra $U_n$ in the variety of left-symmetric algebras is a homogeneous quadratic algebra generated by $2n$ elements $l_1,\ldots,l_n,r_1,\ldots,r_n$, which contains both the polynomial algebra $L_n=K[l_1,\ldots,l_n]$ and the free associative algebra $R_n=K\langle r_1,\ldots,r_n\rangle$. We show that the automorphism groups of the polynomial algebra $L_n$ and the algebra $U_n$ are isomorphic for all $n\geq 2$, based on a detailed analysis of locally nilpotent derivations. In contrast, we show that this isomorphism does not hold for $n=1$, and we provide a complete description of all automorphisms and locally nilpotent derivations of $U_1$. |
| title | Automorphisms and derivations of a universal left-symmetric enveloping algebra |
| topic | Rings and Algebras 13F20, 14R10, 16D25, 16W20, 17D25 |
| url | https://arxiv.org/abs/2506.24051 |