Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: The higher order $q$-Dolan-Grady relations and quantum integrable
  systems   In this thesis, the connection between recently introduced algebraic
structures (tridiagonal algebra, $q$-Onsager algebra, generalized $q-$Onsager
algebras), related representation theory (tridiagonal pair, Leonard pair,
orthogonal polynomials), some properties of these algebras and the analysis of
related quantum integrable models on the lattice (the $XXZ$ open spin chain at
roots of unity) is first reviewed. Then, the main results of the thesis are
described: (i) for the class of $q-$Onsager algebras associated with
$\widehat{sl_2}$ and ADE type simply-laced affine Lie algebras, higher order
analogs of Lusztig's relations are conjectured and proven in various cases,
(ii) for the open $XXZ$ spin chain at roots of unity, new elements (that are
divided polynomials of $q-$Onsager generators) are introduced and some of their
properties are studied. These two elements together with the two basic elements
of the $q-$Onsager algebra generate a new algebra, which can be understood as
an analog of Lusztig's quantum group for the $q-$Onsager algebra. Some
perspectives are presented.

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The categories of this paper are: hep-th
28
The categories of this paper are: alg-geom
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The categories of this paper are: funct-an
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The categories of this paper are: dg-ga
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The categories of this paper are: plasm-ph
13
The categories of this paper are: quant-ph
37
The categories of this paper are: supr-con
0
The categories of this paper are: hep-ph
36
The categories of this paper are: cmp-lg
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The categories of this paper are: nlin
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The categories of this paper are: adap-org
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The categories of this paper are: cond-mat
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The categories of this paper are: mtrl-th
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The categories of this paper are: comp-gas
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The categories of this paper are: atom-ph
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The categories of this paper are: econ