Heun algebras of Lie type

Abstract

We introduce Heun algebras of Lie type. They are obtained from bispectral pairs associated to simple or solvable Lie algebras of dimension three or four. For su(2), this leads to the Heun-Krawtchouk algebra. The corresponding Heun-Krawtchouk operator is identified as the Hamiltonian of the quantum analogue of the Zhukovski-Voltera gyrostat. For su(1, 1), one obtains the Heun algebras attached to the Meixner, Meixner-Pollaczek and Laguerre polynomials. These Heun algebras are shown to be isomorphic the the Hahn algebra. Focusing on the harmonic oscillator algebra ho leads to the Heun-Charlier algebra. The connections to orthogonal polynomials are achieved through realizations of the underlying Lie algebras in terms of difference and differential operators. In the su(1, 1) cases, it is observed that the Heun operator can be transformed into the Hahn, Continuous Hahn and Confluent Heun operators respectively.

The missing label of $\mathfrak{su}_3$ and its symmetry

“…of relation ( 27a) is usually called a algebraic Heun-Hahn operator. The notion of algebraic Heun operator which generalizes the standard Heun operator were recently introduced in [17] and has been studied in different cases [26,2,11,4].…”

The rational Sklyanin algebra and the Wilson and para-Racah polynomials

“…These algebras have been the object of much attention from the perspective of algebraic geometry [28,34,18]. Classes of Heun operators can be defined [17] from the property that they increase by no more than one the degree of polynomials defined on certain continuous or discrete domains; they have been the focus of a continued research effort [29,2,33,10,30,3,5] with many applications [26,21,7,8,9,4,1]. A key observation for our purposes is that a special category of these operators, referred to as S-Heun operators, offers a path towards the identification of interesting Sklyanin-like algebras through the relations they realize.…”

The Heun-Racah and Heun-Bannai-Ito algebras

“…To do so, one considers the algebra generated by such a pair and defines the algebraic Heun operator as the generic bilinear combination of the operators in the bispectral pair. Such a construction has proved useful [9] in the theory of time and band limiting [23,21] as well as in the study of entanglement in fermionic chains [7,8] and the related algebraic structures are now being studied. Of particular interest are the algebraic Heun operators constructed from the bispectral problems arising in the theory of orthogonal polynomials.…”

“…In general, one is lead to the study of algebras, referred to in terms of the underlying polynomials, for instance, if X and Y are the generators of the Racah algebra, the algebra generated by the pair X, W or Y, W shall be called the Heun-Racah algebra. Characterizations of these Heun algebras have been done in [3] for the Heun-Askey-Wilson algebra, in [27] for the Heun-Hahn algebra and in [9] for the Heun algebras of the Lie type which encompasses the cases of the Krawtchouk, Meixner, Meixner-Pollaczek, Laguerre and Charlier polynomials. In this paper, a similar characterization is made of the Heun-Racah and the Heun-Bannai-Ito algebras.…”