Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Embedding of the Racah Algebra R($\boldsymbol{n}$) and
  Superintegrability   The rank-$1$ Racah algebra $R(3)$ plays a pivotal role in the theory of
superintegrable systems. It appears as the symmetry algebra of the
$3$-parameter system on the $2$-sphere from which all second-order conformally
flat superintegrable models in $2$D can be obtained by means of suitable limits
and contractions. A higher rank generalization of $R(3)$, the so-called rank
$n-2$ Racah algebra $R(n)$, has been considered recently and showed to be the
symmetry algebra of the general superintegrable model on the $(n-1)$-sphere. In
the present work, we show that such an algebraic structure naturally arises as
embedded inside a larger quadratic algebra characterizing $n$D superintegrable
models with non-central terms. This is shown both in classical and quantum
mechanics through suitable (symplectic or differential) realisations of the
Racah and additional generators. Among the main results, we present an explicit
construction of the complete symmetry algebras for two families of
$n$-dimensional maximally superintegrable models, the Smorodinsky-Winternitz
system and the generalized Kepler-Coulomb system. For both families, the
underlying symmetry algebras are higher-rank quadratic algebras containing the
Racah algebra $R(n)$ as subalgebra. High-order algebraic relations among the
generators of the full quadratic algebras are also obtained both in the
classical and quantum frameworks. These results should shed new light to the
further understanding of the structures of quadratic algebras in the context of
superintegrable systems.

16
The categories of this paper are: stat
8
The categories of this paper are: hep-ex
33
The categories of this paper are: plasm-ph
6
The categories of this paper are: astro-ph
36
The categories of this paper are: cmp-lg
7
The categories of this paper are: hep-th
11
The categories of this paper are: hep-lat
4
The categories of this paper are: cond-mat
34
The categories of this paper are: chem-ph
28
The categories of this paper are: alg-geom
10
The categories of this paper are: q-bio
31
The categories of this paper are: ao-sci
30
The categories of this paper are: q-alg
35
The categories of this paper are: mtrl-th
12
The categories of this paper are: nucl-th
18
The categories of this paper are: econ