Smooth geometry of skew PBW extensions over commutative polynomial rings I
Andrés Rubiano, Armando Reyes
TL;DR
This work advances noncommutative differential geometry for SPBW extensions by establishing differential smoothness over commutative polynomial rings in one and two indeterminates, and then extending to two, three, and general indeterminates in both $oldsymbol{k[t]}$ and $oldsymbol{L_g}$ settings. The authors construct explicit differential calculi of dimension matching the GK-dimension (e.g., 3, 4, and in general $n+1$), using carefully designed automorphisms $ u_t$ and $ u_{x_i}$ to ensure Leibniz compatibility and integrability with volume forms. They provide a comprehensive framework of sufficient conditions under which SPBW extensions are differentially smooth, including detailed treatment of two-, three-, and $n$-indeterminate cases, and they outline analogous results for Lg-indeterminates. The results contribute to the noncommutative geometry of SPBW extensions and suggest avenues for further exploration in automorphism theory and higher-dimensional calculi, with potential connections to other PBW-type algebras. Future work will tackle broader indeterminates and the tame/wild automorphism problem in this algebraic-geometric context, sharpening the differential-smoothness program for SPBW extensions.
Abstract
In this paper, we investigate the differential smoothness of skew PBW extensions over commutative polynomial rings on one and two indeterminates.