Schott, René; Staples, G. Stacey Operator homology and cohomology in Clifford algebras. (English) Zbl 1217.15035 Cubo 12, No. 2, 299-326 (2010). The authors consider canonical raising and lowering operators defined on a Clifford algebra of arbitrary signature. These operators are used to define chains and cochains of vector spaces underlying the Clifford algebra, to compute the associated homology and cohomology, and to derive long exact sequences of underlying vector spaces. The vector spaces appearing in the chains and cochains correspond to the Appell system decomposition of the Clifford algebra. Kernels of lowering operators \(\nabla \) and raising operators \(\mathcal{R}\) are explicitly computed, giving solutions to equations \(\nabla x=0\) and \(\mathcal{R}x=0\). Connections with quantum probability and graphical interpretation of lowering and raising operators are discussed. Reviewer: Georgi Hristov Georgiev (Shumen) Cited in 3 Documents MSC: 15A66 Clifford algebras, spinors 60B99 Probability theory on algebraic and topological structures 81P15 Quantum measurement theory, state operations, state preparations 55N99 Homology and cohomology theories in algebraic topology Keywords:operator calculus; Clifford algebras; Appell systems; homology; cohomology; quantum probability Software:CliffOC PDFBibTeX XMLCite \textit{R. Schott} and \textit{G. S. Staples}, Cubo 12, No. 2, 299--326 (2010; Zbl 1217.15035) Full Text: DOI