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

In this paper, the two-parameter quantum Grassmann algebra ωr,s(n) is constructed using the two-parameter quantum divided power algebra r,s(n) and the two-parameter quantum exterior algebra δr,s(n). By defining two special chiral (r,s)-quantum partial operators i, i′ over them (where = n, or n), ωr,s(n) is shown to be a Ur,s()-module algebra. The two-parameter quantum de Rham complex (ωr,s(n),d•), as well as the truncated subcomplexes (ωr,s(n,m),d•) in the root of unity case, are also constructed through defining the compatible (r,s)-differentials dk, which are proven to be U-module homomorphisms (where U = Ur,s() or r,s()). 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 (ωr,s(n),d•) by virtue of a "modular"trick.