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Journal of Mathematical Physics / Volume 46 / Issue 6 / METHODS OF MATHEMATICAL PHYSICS

J. Math. Phys. 46, 063503 (2005); http://dx.doi.org/10.1063/1.1915291 (5 pages)

On the Casimir of the group ISL(n,R) and its algebraic decomposition

J. N. Pecina-Cruz

Bureau of Economic Geology, J.J. Pickle Research Campus, The University of Texas at Austin, University Station, Box X, Austin, Texas 78713-8294

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(Received 2 March 2005; accepted 18 March 2005; published online 16 May 2005)

In this paper, an explicit expression for the Casimir operator (or the Casimir invariant) of the inhomogeneous group ISL(n,R) in its enveloping algebra is proposed, which using contractions of the tensorial indices of the generating operators Pρ and Eμν may be presented in the following [slightly more comprehensible as Eq. ( 1 )] form. The Casimir is obtained by symmetrizing this expression. This tensor form is useful in the classification of particles in affine gravitational gauge theories; such as that based on ISL(4,R). It is also proven that the Casimir of ISL(n,R) can be decomposed in terms of the Casimirs of its little groups, a key point in the posterior construction of its irreducible representations.

© 2005 American Institute of Physics

Article Outline

  1. INTRODUCTION
  2. CONSTRUCTION OF THE FORMULA FOR THE CASIMIR OF ISL(n,R)
  3. THE INDUCTION PROOF
  4. THE ALGEBRAIC DECOMPOSITION OF ISL(N,R)
  5. CONCLUSION

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KEYWORDS and PACS

Keywords

gauge field theory, group theory, quantum gravity, tensors

PACS

ARTICLE DATA

Digital Object Identifier
http://dx.doi.org/10.1063/1.1915291

PUBLICATION DATA

ISSN

0022-2488 (print)  
1089-7658 (online)

Publisher

$abstract.society.value is a Member of CrossRef American Institute of Physics

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