Hidden supersymmetry and quadratic deformations of the space-time conformal superalgebra

Abstract

We analyze the structure of the family of quadratic superalgebras, introduced in J Phys A 44(23):235205 (2011), for the quadratic deformations of N = 1 space-time conformal supersymmetry. We characterize in particular the 'zero-step' modules for this case. In such modules, the odd generators vanish identically, and the quadratic superalgebra is realized on a single irreducible representation of the even subalgebra (which is a Lie algebra). In the case under study, the quadratic deformations of N = 1 space-time conformal supersymmetry, it is shown that each massless positive energy unitary irreducible representation (in the standard classification of Mack), forms such a zero-step module, for an appropriate parameter choice amongst the quadratic family (with vanishing central charge). For these massless particle multiplets therefore, quadratic supersymmetry is unbroken, in that the supersymmetry generators annihilate all physical states (including the vacuum state), while at the same time, superpartners do not exist.S 2 := {(i, j)|i, j ∈ S 1 , x i x j is not a leading monomial}.

Quantum superintegrable system with a novel chain structure of quadratic algebras

“…The sub-structures discovered will allow further application of algebraic method for higher rank polynomial algebras and corresponding superintegrable models. Quadratic deformation of Lie super algebras have been observed in other context [21].…”

Three-dimensional gravity as a noncommutative gauge theory

“…Another possible direction of further investigation is to consider our construction embedded in the Lie superalgebra of SU(2,2/1) which is isomorphic to the algebra of the superconformal spacetime symmetry group [88]. Gauging of the latter leads to N = 1 conformal supergravity [8,7,89,90,91].…”

Quadratic algebras as commutants of algebraic Hamiltonians in the enveloping algebra of Schrödinger algebras

“…It follows at once from these commutators that the polynomials {M 1 , M 3 − M 4 , M 5 } do no more generate a quadratic algebra, but a solvable Lie algebra isomorphic to b ⊕ R. 4 3 The Conformal Galilean algebra S(3)…”

“…Now [Z 1 , A 6 ] shows that the polynomial Z 2 = P 2 2 P 2 1 H must also be added to the generators if the algebra is to be quadratic. On the other hand, the commutator [Z 1 , Z 2 ] implies that Z 3 = P 4 2 P 1 P 0 H must also belong to the set of generators. Evaluating recursively the commutators of Z 1 and A 6 with Z k for k ≥ 2, we find that the monomials of the type P α+3β 2 P α 1 P β 0 H with α, β ∈ N must be taken as generators, hence leading to a quadratic algebra that is not finitely generated.…”

“…The classification and hierarchization problem of superintegrable systems has shown that the notion of symmetry can be interpreted far beyond the classical approach using Lie algebras typically appearing in the context of integrals of the motion or spectrum-generating algebras, incorporating other types of algebraic structures such as quadratic algebras [1][2][3]. These algebras appear naturally when analyzing the commutation relations of conserved quantities, and provide an insightful connection with various other mathematical tools, such as the Askey scheme of orthogonal polynomials, the recoupling coefficient formalism, the coalgebra method or the Clebsch-Gordan problem [4,5]. These extended objects, although still incompletely exploited, have been shown to be of crucial importance in order to establish a hierarchy of superintegrable systems and their mutual connection.…”