Automorphisms and derivations of a universal left-symmetric enveloping algebra

TL;DR

This paper investigates the automorphism groups and derivations of a universal left-symmetric algebra $U_n$, revealing an isomorphism with polynomial algebra automorphisms for $n eq 1$ and providing a complete description for $n=1$.

Contribution

It establishes the isomorphism of automorphism groups between $U_n$ and polynomial algebras for all $n eq 1$, and characterizes derivations and automorphisms of $U_1$.

Findings

Automorphism groups of $L_n$ and $U_n$ are isomorphic for all $n eq 1$.

Complete description of automorphisms and derivations of $U_1$.

Differences in automorphism group structure for $n=1$ versus $n eq 1$.

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 $U_1$.