Automorphisms and derivations of a universal left-symmetric enveloping algebra

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 $l_1,\dots ,l_n,r_1,\dots ,r_n$, 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 $n\ge 2$, 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 $U_1$.