Algebraic Structures and Orthogonal Polynomial Theory
Summary
Algebraic structures underpin much of contemporary mathematical physics and pure mathematics, providing the language for symmetries, operator algebras and representation theory. Orthogonal polynomial theory interweaves with these structures by offering explicit bases for modules over associative algebras and by furnishing eigenfunctions for commuting families of difference or differential operators. Central themes include the Askey–Wilson algebra and its degenerations, which encode the bispectrality and duality properties of families such as the q-Askey scheme, and the use of Lie and quantum algebras to generate orthogonality relations, recurrence formulas and connection coefficients. Recent advances have clarified the role of higher-order and ternary algebraic systems in integrable hierarchies, extended the classification of finite-dimensional modules for universal algebras, and demonstrated how multivariate constructs emerge naturally in models of superintegrable systems. These developments enrich approximation theory, spectral analysis and the algebraic approach to nonlinear evolution equations, while also finding applications in quantum computing, signal processing and statistical mechanics.
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Algebraic Structures and Orthogonal Polynomial Theory publication trend
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Technical terms
Algebraic structure: A set with one or more operations (such as addition or multiplication) satisfying specified axioms, for example groups, rings or algebras.
Orthogonal polynomial: A sequence of polynomials in which each is orthogonal to all others with respect to a bilinear form or weight function.
Askey–Wilson algebra: A quadratic algebra generated by two elements whose relations encode the bispectral properties of Askey–Wilson polynomials.
q-Racah polynomial: A family of orthogonal polynomials on a finite set, generalising classical Racah polynomials via a deformation parameter q.
Pöppe triple system: A combinatorial ternary algebraic structure designed to capture integrability properties of non-commutative hierarchies.
Clebsch–Gordan coefficient: A scalar arising in the decomposition of tensor products of representations of a Lie or quantum algebra into irreducible components.
References
- Pöppe triple systems and integrable equations. Partial Differential Equations in Applied Mathematics (2023).
- On finite-dimensional irreducible modules for the universal Askey-Wilson algebra. AIMS Mathematics (2023).
- Coupling coefficients of su q (1,1) and multivariate q-Racah polynomials. Nuclear Physics B (2018).
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