Orthogonalization and polarization of Yangians
Wolfgang Bock, Vyacheslav Futorny, Mikhail Neklyudov, Jian Zhang
TL;DR
This work introduces two complementary realizations of Yangian-type algebras related to ${\mathfrak{gl}}_n$: a polynomial-recurrence realization $OY({\mathfrak{gl}}_n, a,b)$ built from a sequence of polynomials $p_m$ and an associated generating function, and a polarization construction yielding a ternary Yangian $Yt(\mathcal{A})$ that is a flat deformation of $U(\mathcal{A}^{tern}[x])$. The authors develop a quantum Christoffel-Darboux formula in the setting of two copies of the orthogonal-polynomial realization and analyze the case of Dickson polynomials, which reduces to an inhomogeneous RTT framework with a quantum determinant $qdet\tilde{T}(u)$ that centralizes the algebra. They further illustrate the framework with Hermite, non-orthogonal, $q$-Pochhammer, infinite Pochhammer, and Bessel examples, deriving corresponding RTT- or YBE-type relations and illustrating the flexibility of the approach. The polarization construction introduces a ternary extension of Lie algebras and a corresponding ternary Yangian, including an evaluation map to $U(\mathcal{A}^{tern})$, thereby linking Yangians to ternary algebra structures and broadening potential applications in quantum integrable systems and representation theory.
Abstract
For every family of orthogonal polynomials, we define a new realization of the Yangian of ${\mathfrak{gl}}_n$. Except in the case of Dickson polynomials, the new realizations do not satisfy the RTT relation. We obtain an analogue of the Christoffel-Darboux formula. Similar construction can be made for any family of functions satisfying certain recurrence relations, for example, $q$-Pochhhammer symbols and Bessel functions. Furthermore, using an analogue of the Jordan-Schwinger map, we define the ternary Yangian for a Lie algebra as a flat deformation of the current algebra of certain ternary extension of the given Lie algebra.