2025/02/06 by Ge Feng, Naihong Hu
Computer Science · Mathematics · #Advanced Topics in Algebra #Algebra over a field #Algebraic structures and combinatorial models #Chern–Weil homomorphism #Cohomology #De Rham cohomology #Equivariant cohomology #Mathematics #Physics #Pure mathematics #Quantum #Quantum Computing Algorithms and Architecture #Quantum group #Quantum mechanics
paper · doi:10.1142/s0219498826501574
openalex publication_date 2025/02/06 · crossref created 2025/02/06 · crossref issued 2025/03/13 · crossref published 2025/03/13 · crossref published-online 2025/03/13 · openalex created_date 2025/10/10 · crossref deposited 2026/08/03 · crossref indexed 2026/08/03 · openalex updated_date 2026/08/05 · crossref published-print 2026/10/01
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 U-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.