Operator Homology and Cohomology in Clifford Algebras

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...

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Autores principales:	Schott,René, Staples,G. Stacey
Lenguaje:	English
Publicado:	Universidad de La Frontera. Departamento de Matemática y Estadística. 2010
Materias:	Operator calculus Clifford algebras Appell systems quantum probability homology cohomology Fock space fermion
Acceso en línea:	http://www.scielo.cl/scielo.php?script=sci_arttext&pid=S0719-06462010000200018
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spelling	oai:scielo:S0719-064620100002000182018-10-08Operator Homology and Cohomology in Clifford AlgebrasSchott,RenéStaples,G. Stacey Operator calculus Clifford algebras Appell systems quantum probability homology cohomology Fock space fermion 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.info:eu-repo/semantics/openAccessUniversidad de La Frontera. Departamento de Matemática y Estadística.Cubo (Temuco) v.12 n.2 20102010-01-01text/htmlhttp://www.scielo.cl/scielo.php?script=sci_arttext&pid=S0719-06462010000200018en10.4067/S0719-06462010000200018
institution	Scielo Chile
collection	Scielo Chile
language	English
topic	Operator calculus Clifford algebras Appell systems quantum probability homology cohomology Fock space fermion
spellingShingle	Operator calculus Clifford algebras Appell systems quantum probability homology cohomology Fock space fermion Schott,René Staples,G. Stacey Operator Homology and Cohomology in Clifford Algebras
description	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.
author	Schott,René Staples,G. Stacey
author_facet	Schott,René Staples,G. Stacey
author_sort	Schott,René
title	Operator Homology and Cohomology in Clifford Algebras
title_short	Operator Homology and Cohomology in Clifford Algebras
title_full	Operator Homology and Cohomology in Clifford Algebras
title_fullStr	Operator Homology and Cohomology in Clifford Algebras
title_full_unstemmed	Operator Homology and Cohomology in Clifford Algebras
title_sort	operator homology and cohomology in clifford algebras
publisher	Universidad de La Frontera. Departamento de Matemática y Estadística.
publishDate	2010
url	http://www.scielo.cl/scielo.php?script=sci_arttext&pid=S0719-06462010000200018
work_keys_str_mv	AT schottrene operatorhomologyandcohomologyincliffordalgebras AT staplesgstacey operatorhomologyandcohomologyincliffordalgebras
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Operator Homology and Cohomology in Clifford Algebras

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