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Operator homology and cohomology in Clifford algebras. (English) Zbl 1217.15035

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.

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

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