CUBO A Mathematical Journal Vol.12, N°02, (299–326). June 2010

 

Operator Homology and Cohomology in Clifford Algebras

 

René Schott and G. Stacey Staples

IECN and LORIA, Université Henri Poincaré-Nancy I, BP 239, 54506 Vandoeuvre-lés-Nancy, France email: schott@loria.fr

Department of Mathematics and Statistics, Southern Illinois University Edwardsville, Edwardsville, IL 62026-1653,USA email: sstaple@siue.edu

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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 Rx = 0. Connections with quantum probability and graphical interpretations of the lowering and raising operators are discussed.

Key words and phrases: Operator calculus, Clifford algebras, Appell systems, quantum probability, homology, cohomology, Fock space, fermion

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RESUMEN

En recientes trabajos, los autores usaron operadores canónicos de bajada y de elevación para definir sistemas de Appell sobre algebras de Clifford de signo arbitrario. Los sistemas de Appell pueden ser interpretados como soluciones polinomiales de ecuaciones del calor generalizadas, y en teoría de probabilidades estos han sido usados para obtener teoremas de límite no central. La natural malla-descomposición para una algebra de Clifford de signo arbitrario presta una descomposición natural del sistema de Appel. En este trabajo, operadores canónicos de elevación y de bajada definidos sobre una algebra de Clifford de signo arbitrario son usados para definir cadenas y cocadenas de espacios vectoriales de llegada de algebras de Clifford; para calcular los grupos de homología y cohomología asociados; y para obtener el tamaño de las sucesiones exactas de los espacios vectoriales de llegada. Los espacios vectoriales que aparecen en las cadenas y cocadenas corresponden a la descomposición de sistemas de Appell de la algebra de Clifford. Usando MATHEMATICA, son calculados expíıcitamente los núcleos de operadores de bajada ∇ y de operadores de elevación R dando soluciones para las ecuaciones ∇x = 0 y Rx = 0. Son discutidas conecciones con probabilidad cuantica y interpretaciones graficas para los operadores de bajada y de elevación.

AMS Subj. Class.: 15A66, 60B99, 81R05

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References

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Received: January 2009.

Revised: August 2009.