dorsal/arxiv
View SchemaAlgebras, Derivations and Integrals
| Authors | R. Casalbuoni |
|---|---|
| Categories | |
| ArXiv ID | physics/9803024 |
| URL | https://arxiv.org/abs/physics/9803024 |
| DOI | 10.1142/S0217751X98002481 |
Abstract
In the context of the integration over algebras introduced in a previous paper, we obtain several results for a particular class of associative algebras with identity. The algebras of this class are called self-conjugated, and they include, for instance, the paragrassmann algebras of order $p$, the quaternionic algebra and the toroidal algebras. We study the relation between derivations and integration, proving a generalization of the standard result for the Riemann integral about the translational invariance of the measure and the vanishing of the integral of a total derivative (for convenient boundary conditions). We consider also the possibility, given the integration over an algebra, to define from it the integral over a subalgebra, in a way similar to the usual integration over manifolds. That is projecting out the submanifold in the integration measure. We prove that this is possible for paragrassmann algebras of order $p$, once we consider them as subalgebras of the algebra of the $(p+1)\times(p+1)$ matrices. We find also that the integration over the subalgebra coincides with the integral defined in the direct way. As a by-product we can define the integration over a one-dimensional Grassmann algebra as a trace over $2\times 2$ matrices.
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"abstract": "In the context of the integration over algebras introduced in a previous\npaper, we obtain several results for a particular class of associative algebras\nwith identity. The algebras of this class are called self-conjugated, and they\ninclude, for instance, the paragrassmann algebras of order $p$, the\nquaternionic algebra and the toroidal algebras. We study the relation between\nderivations and integration, proving a generalization of the standard result\nfor the Riemann integral about the translational invariance of the measure and\nthe vanishing of the integral of a total derivative (for convenient boundary\nconditions). We consider also the possibility, given the integration over an\nalgebra, to define from it the integral over a subalgebra, in a way similar to\nthe usual integration over manifolds. That is projecting out the submanifold in\nthe integration measure. We prove that this is possible for paragrassmann\nalgebras of order $p$, once we consider them as subalgebras of the algebra of\nthe $(p+1)\\times(p+1)$ matrices. We find also that the integration over the\nsubalgebra coincides with the integral defined in the direct way. As a\nby-product we can define the integration over a one-dimensional Grassmann\nalgebra as a trace over $2\\times 2$ matrices.",
"arxiv_id": "physics/9803024",
"authors": [
"R. Casalbuoni"
],
"categories": [
"math-ph",
"hep-th",
"math.MP",
"math.QA"
],
"doi": "10.1142/S0217751X98002481",
"title": "Algebras, Derivations and Integrals",
"url": "https://arxiv.org/abs/physics/9803024"
},
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