dorsal/arxiv
View SchemaCasimir invariants for the complete family of quasi-simple orthogonal algebras
| Authors | Francisco J. Herranz, Mariano Santander |
|---|---|
| Categories | |
| ArXiv ID | physics/9702032 |
| URL | https://arxiv.org/abs/physics/9702032 |
| DOI | 10.1088/0305-4470/30/15/026 |
| Journal | J.Phys.A30:5411-5426,1997 |
Abstract
A complete choice of generators of the center of the enveloping algebras of real quasi-simple Lie algebras of orthogonal type, for arbitrary dimension, is obtained in a unified setting. The results simultaneously include the well known polynomial invariants of the pseudo-orthogonal algebras $so(p,q)$, as well as the Casimirs for many non-simple algebras such as the inhomogeneous $iso(p,q)$, the Newton-Hooke and Galilei type, etc., which are obtained by contraction(s) starting from the simple algebras $so(p,q)$. The dimension of the center of the enveloping algebra of a quasi-simple orthogonal algebra turns out to be the same as for the simple $so(p,q)$ algebras from which they come by contraction. The structure of the higher order invariants is given in a convenient "pyramidal" manner, in terms of certain sets of "Pauli-Lubanski" elements in the enveloping algebras. As an example showing this approach at work, the scheme is applied to recovering the Casimirs for the (3+1) kinematical algebras. Some prospects on the relevance of these results for the study of expansions are also given.
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"abstract": "A complete choice of generators of the center of the enveloping algebras of\nreal quasi-simple Lie algebras of orthogonal type, for arbitrary dimension, is\nobtained in a unified setting. The results simultaneously include the well\nknown polynomial invariants of the pseudo-orthogonal algebras $so(p,q)$, as\nwell as the Casimirs for many non-simple algebras such as the inhomogeneous\n$iso(p,q)$, the Newton-Hooke and Galilei type, etc., which are obtained by\ncontraction(s) starting from the simple algebras $so(p,q)$. The dimension of\nthe center of the enveloping algebra of a quasi-simple orthogonal algebra turns\nout to be the same as for the simple $so(p,q)$ algebras from which they come by\ncontraction. The structure of the higher order invariants is given in a\nconvenient \"pyramidal\" manner, in terms of certain sets of \"Pauli-Lubanski\"\nelements in the enveloping algebras. As an example showing this approach at\nwork, the scheme is applied to recovering the Casimirs for the (3+1)\nkinematical algebras. Some prospects on the relevance of these results for the\nstudy of expansions are also given.",
"arxiv_id": "physics/9702032",
"authors": [
"Francisco J. Herranz",
"Mariano Santander"
],
"categories": [
"math-ph",
"math.MP"
],
"doi": "10.1088/0305-4470/30/15/026",
"journal_ref": "J.Phys.A30:5411-5426,1997",
"title": "Casimir invariants for the complete family of quasi-simple orthogonal algebras",
"url": "https://arxiv.org/abs/physics/9702032"
},
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