The Racah Algebra and sl n
Hendrik De Bie, Luc Vinet, and Wouter van de Vijver
Abstract We conjecture the existence of an embedding of the Racah algebra into
the universal enveloping algebra of sl n . Evidence of this conjecture is offered by
realizing both algebras using differential operators and giving an embedding in this
realization.
Keywords Racah algebra · Embedding · Lie algebra sl n
1 Introduction
The Racah algebra synthesizes the properties of the Racah polynomials [8, 13],
which are the most complicated univariate discrete orthogonal polynomials in the
Askey scheme [11].
Multivariate Racah polynomials were introduced by Tratnik in [12]. These
polynomials also have a solid algebraic underpinning, as was recently established
in [3] using the higher rank Racah algebra. This higher rank Racah algebra was
initially introduced in [9, 10] in the context of superintegrability and later in [4]
as a subalgebra of intermediate Casimir elements in the n-fold tensor product of
su(1, 1).
Although the initial motivation to introduce the (higher rank) Racah algebra was
to establish a connection with the multivariate Racah polynomials, the algebra has
now become an independent object of study. In particular its relation with other
algebraic structures is part of ongoing investigations. We refer the reader to [1] for
H. De Bie () · W. van de Vijver
Department of Electronics and Information Systems, Faculty of Engineering and Architecture,
Ghent University, Gent, Belgium
e-mail: hendrik.debie@ugent.be; wouter.vandevijver@ugent.be
L. Vinet
Centre de Recherches Mathématiques, Université de Montréal, Montréal, QC, Canada
e-mail: vinet@crm.umontreal.ca
© Springer Nature Switzerland AG 2021
M. B. Paranjape et al. (eds.), Quantum Theory and Symmetries, CRM Series in
Mathematical Physics, https://doi.org/10.1007/978-3-030-55777-5_19
209
Hendrik De Bie, Luc Vinet, and Wouter van de Vijver
Abstract We conjecture the existence of an embedding of the Racah algebra into
the universal enveloping algebra of sl n . Evidence of this conjecture is offered by
realizing both algebras using differential operators and giving an embedding in this
realization.
Keywords Racah algebra · Embedding · Lie algebra sl n
1 Introduction
The Racah algebra synthesizes the properties of the Racah polynomials [8, 13],
which are the most complicated univariate discrete orthogonal polynomials in the
Askey scheme [11].
Multivariate Racah polynomials were introduced by Tratnik in [12]. These
polynomials also have a solid algebraic underpinning, as was recently established
in [3] using the higher rank Racah algebra. This higher rank Racah algebra was
initially introduced in [9, 10] in the context of superintegrability and later in [4]
as a subalgebra of intermediate Casimir elements in the n-fold tensor product of
su(1, 1).
Although the initial motivation to introduce the (higher rank) Racah algebra was
to establish a connection with the multivariate Racah polynomials, the algebra has
now become an independent object of study. In particular its relation with other
algebraic structures is part of ongoing investigations. We refer the reader to [1] for
H. De Bie () · W. van de Vijver
Department of Electronics and Information Systems, Faculty of Engineering and Architecture,
Ghent University, Gent, Belgium
e-mail: hendrik.debie@ugent.be; wouter.vandevijver@ugent.be
L. Vinet
Centre de Recherches Mathématiques, Université de Montréal, Montréal, QC, Canada
e-mail: vinet@crm.umontreal.ca
© Springer Nature Switzerland AG 2021
M. B. Paranjape et al. (eds.), Quantum Theory and Symmetries, CRM Series in
Mathematical Physics, https://doi.org/10.1007/978-3-030-55777-5_19
209