Identifying the $q$-Pochhammer, Dickson, and Bessel algebras of the orthogonalization of Yangians

Bock, Futorny, Neklyudov, and Zhang showed that the Yangian $Y(\mathfrak{gl}_n)$ can be rebuilt from the three-term recurrence of any monic orthogonal-polynomial family: their orthogonalization construction produces an algebra isomorphic to $Y(\mathfrak{gl}_n)$, and applying the same machine to Dickson polynomials (where the resulting algebra $OY_\beta(\mathfrak{gl}_n)\cong Y(\mathfrak{gl}_n)$ carries an inhomogeneous RTT presentation), $q$-Pochhammer symbols, and Bessel-type recurrences produces further associative algebras with explicitly computable defining relations. The paper leaves the nature of these latter algebras undetermined: it is explicitly an open problem whether the $q$-Pochhammer-based algebras of its Section 5.3 are isomorphic to $Y(\mathfrak{gl}_n)$ or define genuinely new quantum algebras, while the Laurent-extension algebra $\widehat{OY}_\beta(\mathfrak{gl}_n)$, the Dickson $k=2$ family, and the Bessel-recurrence algebra are all deferred to future work. This problem asks for the identification and basic structure of all four families.

Background

The Yangian $Y(\mathfrak{gl}_n)$ is the Hopf algebra deformation of the universal enveloping algebra of the polynomial current Lie algebra $\mathfrak{gl}_n[u]$, introduced by Drinfeld in the 1980s; it is the rational $R$-matrix member of the quantum-group family and a symmetry algebra of rational integrable spin chains. In its RTT presentation it is generated by the coefficients $t_{ij}^{(r)}$, $r\ge 0$, of a formal matrix current $T(u)=\sum_{r\ge 0}t_{ij}^{(r)}u^{-r}$ (the formal variable $u$ is the spectral parameter), subject to the single matrix relation $R(u-v)T_1(u)T_2(v)=T_2(v)T_1(u)R(u-v)$, where $R(u-v)=I-P/(u-v)$, $P$ is the permutation operator on $\mathbb{C}^n\otimes\mathbb{C}^n$, and $T_1(u)=T(u)\otimes I$, $T_2(v)=I\otimes T(v)$.

A family of orthogonal polynomials is a sequence $p_r(x)$ with $\deg p_r=r$, orthogonal with respect to a measure; by Favard's theorem this is equivalent to a three-term recurrence $xp_r=p_{r+1}+a_rp_r+b_rp_{r-1}$ with $p_{-1}=0$ (Hermite, Laguerre, and Chebyshev polynomials are classical examples). Bock, Futorny, Neklyudov, and Zhang observed that the Yangian commutation relations can be recast so that orthogonalization of a polynomial family becomes an operation on the algebra itself: for any monic family $\{p_m\}$ with a three-term recurrence they define an associative algebra $OY(\mathfrak{gl}_n,\{p_m\})$ whose relations mirror the recurrence coefficients, and they prove a realization theorem $OY(\mathfrak{gl}_n,\{p_m\})\cong Y(\mathfrak{gl}_n)$, given by a unitriangular change of generators together with a contour-integral kernel $K(z,u)$; the proof uses monicity and the three-term shape in an essential way.

Outside this class the construction produces algebras of unknown identity, and even inside it new presentations raise new structural questions. The Dickson polynomials of the second kind used in the paper (the $k=1$ member of its Section 4 family $d_0^k=2-k$, $d_1^k=x$, $d_{r+1}^k=x\,d_r^k-\beta d_{r-1}^k$) are monic with constant recurrence coefficients, so the algebra $OY_\beta(\mathfrak{gl}_n)$ they generate is still isomorphic to $Y(\mathfrak{gl}_n)$ via the same realization theorem; the new phenomenon is that $OY_\beta(\mathfrak{gl}_n)$ carries an inhomogeneous RTT relation with $R$-matrix $R^\beta(u,v)=I-P/(u+\beta/u-v-\beta/v)$, under which the map $T(u)\mapsto T(\beta/u)$ formally preserves the relations but is not an automorphism of $OY_\beta(\mathfrak{gl}_n)$ because $T(\beta/u)$ is a series in positive powers of $u$; the authors therefore introduce a Laurent-series (completed) extension $\widehat{OY}_\beta(\mathfrak{gl}_n)$ and defer its study. The $q$-Pochhammer symbols $(x;q)_r=\prod_{j=0}^{r-1}(1-xq^j)$, by contrast, fall outside the class: they satisfy the first-order multiplicative recurrence $(x;q)_{r+1}=(1-xq^r)(x;q)_r$ and have non-unit leading coefficient $(-1)^rq^{r(r-1)/2}$; substituting them into the construction gives algebras with generators $m_{ij}^{(r)}=((x;q)_r(E))_{ij}$ (the polynomial $(x;q)_r$ evaluated on the $n\times n$ generator matrix $E$ of the $\mathfrak{gl}_n$-type currents) and defining relations (5.4)-(5.5) that contain diagonal commutator terms of the form $(q^{-r}-q^{-s})[m^{(r)},m^{(s)}]$ and a generating-function form that mixes the generating series $m(u)=\delta+\sum_{r\ge 1}m_{ij}^{(r)}u^{-r}$ evaluated at $qu$ with $m(u)$ itself, i.e. a multiplicative shift of the spectral parameter; the limit $q\to 1$ recovers $Y(\mathfrak{gl}_n)$ via $t_{ij}(u)\mapsto m_{ij}(-u)$. Finally, Bessel-function and other nonstandard recurrences produce further algebras whose properties the paper does not determine. None of the standard realization tools (monic three-term recurrences, unitriangular changes of generators) apply to the $q$-Pochhammer and Bessel families or to the exceptional Dickson type $k=2$, which is precisely why their identity is open.

Problem Statement

Determine the identity and basic structure of the associative algebras produced by the orthogonalization construction of Bock, Futorny, Neklyudov, and Zhang beyond what their Section 2 realization theorem covers. The common objective uniting the requested results is that, for each family below, it be decided whether the resulting algebra is isomorphic to the classical Yangian $Y(\mathfrak{gl}_n)$ (or to some other algebra named explicitly in the answer, via an isomorphism exhibited on generators and verified on the defining relations) or is not isomorphic to $Y(\mathfrak{gl}_n)$, the latter certified by an exhibited structural invariant (for instance a graded dimension or a central element) that separates the algebra from $Y(\mathfrak{gl}_n)$ and so shows it to be genuinely new; together with the structural statements specifically requested. All four families below are required, and resolving a proper subset of the objects they list does not resolve the problem:

1. The $q$-Pochhammer algebras of Section 5.3 — the algebras with generators $m_{ij}^{(r)}=((x;q)_r(E))_{ij}$ — comprise two objects to be decided: the algebra with relations (5.4)-(5.5), and the infinite-Pochhammer algebra with relations (5.8)-(5.9), which adjoins the extra generator $m_{ij}^{(\infty)}$ and is taken for $|q|>1$. The intermediate relations (5.6)-(5.7) present the same algebra as (5.4)-(5.5) in the rescaled generating series $\tilde m^n(u)=(1-u)m(u)/(u;q)_n$ (an invertible change of generators) and its $|q|<1$ limit $n\to\infty$; they are a tool for deciding it, not a further object needing its own identification. The source states: 'It is an open problem to understand if the resulting algebras will be isomorphic to $Y(\mathfrak{gl}_n)$.'
2. The Laurent-extension algebra $\widehat{OY}_\beta(\mathfrak{gl}_n)$ of Section 4: determine its structure — its defining relations, and in particular whether and how the formal involution $T(u)\mapsto T(\beta/u)$ lifts to an automorphism of the completed algebra.
3. The Dickson-polynomial family with $k=2$ — the one Dickson type not covered by the authors' reduction to the second type in Section 4 — which the authors state requires separate treatment: carry out that treatment and identify the resulting algebra.
4. The algebra of Section 5.4 arising from the Bessel-function recurrence: determine its basic structure — its defining relations, and in particular whether it is isomorphic to $Y(\mathfrak{gl}_n)$ or to another algebra named explicitly in the answer.

References

1. Wolfgang Bock, Vyacheslav Futorny, Mikhail Neklyudov, Jian Zhang, 'Orthogonalization and polarization of Yangians', International Journal of Algebra and Computation, published online 2026-05-29, pp. 1-27, DOI 10.1142/S0218196726410092; preprint arXiv:2504.00259 (v1 2025-03-31, v2 2025-11-13), DOI 10.48550/arXiv.2504.00259, https://arxiv.org/abs/2504.00259
2. Nicolas Crampe, Eric Ragoucy, Luc Vinet, Alexei Zhedanov, 'Truncation of the reflection algebra and the Hahn algebra', Journal of Physics A: Mathematical and Theoretical 52 (2019) 35LT01, DOI 10.1088/1751-8121/ab32f8; preprint arXiv:1903.05674, DOI 10.48550/arXiv.1903.05674, https://arxiv.org/abs/1903.05674
3. Felipe Albino dos Santos, Mikhail Neklyudov, Vyacheslav Futorny, 'Superelliptic Affine Lie algebras and orthogonal polynomials', Forum of Mathematics, Sigma 13 (2025) e120, DOI 10.1017/fms.2025.10074; preprint arXiv:2402.02947, DOI 10.48550/arXiv.2402.02947, https://arxiv.org/abs/2402.02947
4. Felipe Albino dos Santos, Mikhail Neklyudov, Vyacheslav Futorny, 'Superelliptic Affine Lie algebras and orthogonal polynomials II', preprint arXiv:2603.29082 (2026), DOI 10.48550/arXiv.2603.29082, https://arxiv.org/abs/2603.29082