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On the algebra of symmetries of Laplace and Dirac operators

Hendrik De Bie (UGent) , Roy Oste (UGent) and Joris Van der Jeugt (UGent)
(2018) LETTERS IN MATHEMATICAL PHYSICS. 108(8). p.1905-1953
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Abstract
We consider a generalization of the classical Laplace operator, which includes the Laplace-Dunkl operator defined in terms of the differential-difference operators associated with finite reflection groups called Dunkl operators. For this Laplace-like operator, we determine a set of symmetries commuting with it, in the form of generalized angular momentum operators, and we present the algebraic relations for the symmetry algebra. In this context, the generalized Dirac operator is then defined as a square root of our Laplace-like operator. We explicitly determine a family of graded operators which commute or anticommute with our Dirac-like operator depending on their degree. The algebra generated by these symmetry operators is shown to be a generalization of the standard angular momentum algebra and the recently defined higher-rank Bannai-Ito algebra.
Keywords
BANNAI-ITO ALGEBRA, DUNKL OPERATORS, EQUATIONS, Laplace operator, Dirac operator, Dunkl operator, Symmetry algebra, Bannai-Ito algebra

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Chicago
De Bie, Hendrik, Roy Oste, and Joris Van der Jeugt. 2018. “On the Algebra of Symmetries of Laplace and Dirac Operators.” Letters in Mathematical Physics 108 (8): 1905–1953.
APA
De Bie, H., Oste, R., & Van der Jeugt, J. (2018). On the algebra of symmetries of Laplace and Dirac operators. LETTERS IN MATHEMATICAL PHYSICS, 108(8), 1905–1953.
Vancouver
1.
De Bie H, Oste R, Van der Jeugt J. On the algebra of symmetries of Laplace and Dirac operators. LETTERS IN MATHEMATICAL PHYSICS. 2018;108(8):1905–53.
MLA
De Bie, Hendrik, Roy Oste, and Joris Van der Jeugt. “On the Algebra of Symmetries of Laplace and Dirac Operators.” LETTERS IN MATHEMATICAL PHYSICS 108.8 (2018): 1905–1953. Print.
@article{8567845,
  abstract     = {We consider a generalization of the classical Laplace operator, which includes the Laplace-Dunkl operator defined in terms of the differential-difference operators associated with finite reflection groups called Dunkl operators. For this Laplace-like operator, we determine a set of symmetries commuting with it, in the form of generalized angular momentum operators, and we present the algebraic relations for the symmetry algebra. In this context, the generalized Dirac operator is then defined as a square root of our Laplace-like operator. We explicitly determine a family of graded operators which commute or anticommute with our Dirac-like operator depending on their degree. The algebra generated by these symmetry operators is shown to be a generalization of the standard angular momentum algebra and the recently defined higher-rank Bannai-Ito algebra.},
  author       = {De Bie, Hendrik and Oste, Roy and Van der Jeugt, Joris},
  issn         = {0377-9017},
  journal      = {LETTERS IN MATHEMATICAL PHYSICS},
  language     = {eng},
  number       = {8},
  pages        = {1905--1953},
  title        = {On the algebra of symmetries of Laplace and Dirac operators},
  url          = {http://dx.doi.org/10.1007/s11005-018-1065-0},
  volume       = {108},
  year         = {2018},
}

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