Three dimensional quadratic algebras : Some realizations and representations
نویسندگان
چکیده
Four classes of three dimensional quadratic algebras of the type [Q0, Q±] = ±Q±, [Q+, Q−] = aQ0 + bQ0 + c, where (a, b, c) are constants or central elements of the algebra, are constructed using a generalization of the well known two-mode bosonic realizations of su(2) and su(1, 1). The resulting matrix representations and single variable differential operator realizations are obtained. Some remarks on the mathematical and physical relevance of such algebras are given.
منابع مشابه
/ 98 03 25 3 v 2 1 7 D ec 1 99 8 On realizations of “ nonlinear ” Lie algebras by differential operators
We study realizations of polynomial deformations of the sl(2, ℜ)-Lie algebra in terms of differential operators strongly related to bosonic operators. We also distinguish their finite-and infinite-dimensional representations. The linear, quadratic and cubic cases are explicitly visited but the method works for arbitrary degrees in the polynomial functions. Multi-boson Hamiltonians are studied i...
متن کاملar X iv : h ep - t h / 98 03 25 3 v 1 3 1 M ar 1 99 8 On realizations of nonlinear Lie algebras by differential operators and some physical applications
We study realizations of polynomial deformations of the sl(2, ℜ)-Lie algebra in terms of differential operators strongly related to bosonic operators. We also distinguish their finite-and infinite-dimensional representations. The linear, quadratic and cubic cases are explicitly visited but the method works for arbitrary degrees in the polynomial functions. Specific physical Hamiltonians are obt...
متن کاملRealizations of Indecomposable Solvable 4-Dimensional Real Lie Algebras
Inequivalent twoand three-dimensional Lie algebras were classified in XIX century by Lie [1]. In 1963 Mubaraksyanov classified threeand four-dimensional real Lie algebras [2] (see also those results in Patera and Winternitz [3]). In 1989 Mahomed and Leach obtained realizations of threedimensional Lie algebras in terms of vector fields defined on the plane [4]. Mahomed and Soh tried to obtain re...
متن کاملPolynomial Algebras and Their Applications
A way to construct and classify the three dimensional polynomially deformed algebras is given and the irreducible representations is presented. for the quadratic algebras 4 different algebras are obtained and for cubic algebras 12 different classes are constructed. Applications to quantum mechanical systems including supersymmetric quantum mechanics are discussed
متن کاملQuantum Super-Integrable Systems as Exactly Solvable Models
We consider some examples of quantum super-integrable systems and the associated nonlinear extensions of Lie algebras. The intimate relationship between superintegrability and exact solvability is illustrated. Eigenfunctions are constructed through the action of the commuting operators. Finite dimensional representations of the quadratic algebras are thus constructed in a way analogous to that ...
متن کامل