Clifford Operator Calculus: Homology, Cohomology, and Appell Systems

Appell systems can be interpreted as polynomial solutions of generalized heat equations, and in probability theory they have been used to obtain non-central limit theorems. The natural grade-decomposition of a Clifford algebra of arbitrary signature lends it a natural Appell system decomposition. Motivated by evolution equations on Clifford algebras, combinatorial raising and lowering operators defined on Clifford algebras lead to a theory of invertible leftand rightClifford Appell systems. These operators further allow one to define chains and cochains of vector spaces corresponding to the algebra’s Appell system decomposition. The associated homology and cohomology groups can then be computed, and long exact sequences of underlying vector spaces can be derived. Operator calculus methods also readily lend themselves to convenient symbolic computations; for example, kernels of lowering and raising operators can be explicitly computed with Mathematica.