Quadratic algebras as commutants of algebraic Hamiltonians in the enveloping algebra of Schrödinger algebras
We discuss a procedure to determine finite sets M within the commutant of an algebraic Hamiltonian in the enveloping algebra of a Lie algebra g such that their generators define a quadratic algebra. Although independent from any realization of Lie algebras by differential operators, the method is pa...
| Authors: | , |
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| Format: | article |
| Publication Date: | 2022 |
| Country: | España |
| Institution: | Universidad Complutense de Madrid (UCM) |
| Repository: | Docta Complutense |
| Language: | English |
| OAI Identifier: | oai:docta.ucm.es:20.500.14352/71836 |
| Online Access: | https://hdl.handle.net/20.500.14352/71836 |
| Access Level: | Open access |
| Keyword: | 512.554.3 Quadratic algebras Superintegrable systems Racah algebras Conserved quantities Quantum Hamiltonian Álgebra 1201 Álgebra |
| Summary: | We discuss a procedure to determine finite sets M within the commutant of an algebraic Hamiltonian in the enveloping algebra of a Lie algebra g such that their generators define a quadratic algebra. Although independent from any realization of Lie algebras by differential operators, the method is partially based on an analytical approach, and uses the coadjoint representation of the Lie algebra g. The procedure, valid for non-semisimple algebras, is tested for the centrally extended Schrödinger algebras Ŝ(n) for various different choices of algebraic Hamiltonian. For the so-called extended Cartan solvable case, it is shown how the existence of minimal quadratic algebras can be inferred without explicitly manipulating the enveloping algebra. |
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