Automorphisms and derivations of a universal left-symmetric enveloping algebra

D. Zhangazinova; A. Naurazbekova; U. Umirbaev

paper

Automorphisms and derivations of a universal left-symmetric enveloping algebra

arXiv:2506.24051

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 $2 n$ elements $l_{1}, \dots, l_{n}, r_{1}, \dots, r_{n}$ , which contains both the polynomial algebra $L_{n} = K [l_{1}, \dots, l_{n}]$ and the free associative algebra $R_{n} = K ⟨ r_{1}, \dots, r_{n} ⟩$ . 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 $U_{1}$ .

24 pages