Canonical deformations of de Rham complexes
Let \(\Omega^\bullet_{A/k}\) be the de Rham complex of the commutative polynomial ring \(A=k[x_1,\dots,x_n]\) relative to a field \(k\). The author investigates a special deformation of \(\Omega^\bullet_{A/k}\) in the category of differential graded algebras that appeared in the paper of \textit{S. Woronowicz} [Commun. Math. Phys. 122, No. 1, 125-170 (1989; Zbl 0751.58042)]. In fact, such a deformation \(\Omega^\bullet_q\) can be considered as an analog of de Rham complex of the non-commutative multi-parameter quantum deformation \(A_q\) of \(A\) in the sense of \textit{M. Artin} et al. [Commun. Pure Appl. Math. 44, No. 8/9, 879-895 (1991; Zbl 0753.17015)]. Using the formalism of duality developed by the author [J. Algebra 153, No. 1, 41-84 (1992; Zbl 0790.18005)], he describes an integration over \(A_q\) by means of a canonical isomorphism \(\int:\text{Ext}^n_{A_q}(k,\Omega^n_q)\to k\) induced by the natural extension of the standard contraction of differential forms along the Euler vector field of the de Rham complex \(\Omega^\bullet_{A/k}\) to its deformation \(\Omega^\bullet_q\). More generally, similar results hold in the case where \(A\) is the skew homogeneous coordinate ring.
categories of differential graded algebras
non-commutative multi-parameter quantum deformations
scientific article; zbMATH DE number 848988