Automorphisms and derivations of a universal left-symmetric enveloping algebra

D. Zhangazinova; A. Naurazbekova; U. Umirbaev

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arXiv · 2506.24051

Automorphisms and derivations of a universal left-symmetric enveloping algebra

D. Zhangazinova·A. Naurazbekova·U. Umirbaev

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Abstract

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 $U_1$.

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D. Zhangazinova, A. Naurazbekova, U. Umirbaev. 2025-06-30. Automorphisms and derivations of a universal left-symmetric enveloping algebra. https://arxiv.org/abs/2506.24051

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