Operator homology and cohomology in Clifford algebras

Abstract

In recent work, the authors used canonical lowering and raising operators to define Appell systems on Clifford algebras of arbitrary signature. 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. In the current work, canonical raising and lowering operators defined on a Clifford algebra of arbitrary signature are used to define chains and cochains of vector spaces underlying the Clifford algebra, to compute the associated homology and cohomology groups, 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. Using Mathematica, kernels of lowering operators ∇ and raising operators ℛ are explicitly computed, giving solutions to equations ∇ x = 0 and ℛ x = 0. Connections with quantum probability and graphical interpretations of the lowering and raising operators are discussed.