Homological Dimensions of the Algebra Formed by Entire Functions of Elements of a Nilpotent Lie Algebra

This note deals with homological characteristics of algebras of holomorphic functions of noncommuting variables generated by a finite-dimensional nilpotent Lie algebra $\mathfrak{g}$. It is proved that the embedding $\mathcal{U}(\mathfrak{g})\to\mathcal{O}_{\mathfrak{g}}$ of the universal enveloping algebra $\mathcal{U}(\mathfrak{g})$ of $\mathfrak{g}$ into its Arens–Michael hull $\mathcal{O}_{\mathfrak{g}}$ is an absolute localization in the sense of Taylor provided that $[\mathfrak{g},[\mathfrak{g},\mathfrak{g}]]=0$.