Two-parameter quantum de Rham cohomology over two-parameter quantum divided power algebra

In this paper, the two-parameter quantum Grassmann algebra [Formula: see text] is constructed using the two-parameter quantum divided power algebra [Formula: see text] and the two-parameter quantum exterior algebra [Formula: see text]. By defining two special chiral [Formula: see text]-quantum partial operators [Formula: see text] over them (where [Formula: see text], or [Formula: see text], [Formula: see text] is shown to be a [Formula: see text]-module algebra. The two-parameter quantum de Rham complex [Formula: see text], as well as the truncated subcomplexes [Formula: see text] in the root of unity case, are also constructed through defining the compatible [Formula: see text]-differentials [Formula: see text], which are proven to be [Formula: see text]-module homomorphisms (where [Formula: see text] or [Formula: see text]). For the latter, the corresponding two-parameter quantum de Rham cohomologies, together with their dimensions are determined. Based on it, the Poincaré lemma is shown to hold for the non-truncated case [Formula: see text] by virtue of a “modular” trick.