University of Tasmania
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Quadratic superalgebras in mathematics and physics

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posted on 2023-05-28, 10:13 authored by Yates, LA
We introduce in this thesis a class of quadratic deformations of Lie superalgebras which we term quadratic superalgebras. These are finitely-generated algebras with a `Z`\\(_{_2}\\)-graded structure comprising an even and an odd part; the even part is an ordinary Lie algebra, the odd part is a module of the even part, and the anticommutator of two odd elements closes quadratically on the even generators. One motivation to study algebras with these structural properties is their arising in the observable algebra of gauge invariant fields in Hamiltonian lattice QCD [44] and the subsequent study of polynomial gl(n) superalgebras [45]. The present work both broadens the scope and extends the analysis of the latter. For this class of algebras we derive a Poincare-Birkhoff-Witt theorem, including an explicit ordered basis, which we then employ as a means to investigate the structure of irreducible modules; these are analogous to Kac modules for Lie superalgebras. Further rationale to study quadratic superalgebras is due to the remarkable existence of zero-step modules; these are so-called atypical modules for which the entire irreducible module of the quadratic superalgebra consists of a single irreducible module of the even subalgebra. In addition to their mathematical aspects, we investigate in this thesis an application of quadratic superalgebras in the context of space-time conformal supersymmetry. We show that the algebra of N = 1 space-time conformal supersymmetry, su(2; 2=1), arises as a contraction limit of a certain quadratic superalgebra. In this setting we exploit the existence of zero-step modules which, for a fixed parameter choice of the quadratic family under consideration, coincide with the massless positive energy unitary irreducible representations (in the standard classifcation of Mack) of the even subalgebra. For these massless particle multiplets the odd generators vanish identically and supersymmetry is carried (unbroken) without the accompaniment of superpartners. Thus, in the context of extended non-linear symmetry principles and their role in determining the spectrum of fundamental particles, we point out that there exist candidate algebraic structures which implement (extended) supersymmetric invariance while at the same time obviating the need for every particle of the standard model to be accompanied by a superpartner.


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Copyright 2018 the author The key results in chapter 2 and 5 have been published in: Jarvis, P. D., Rudolph, G., Yates, L. A., 2011. A class of quadratic deformations of Lie superalgebras, Journal of physics A: Mathematical and theoretical 44, 23, 235205, 1-24 The key results in chapter 5 and 6 have been published in: Yates, L. A., Jarvis, P. D., 2017. Hidden supersymmetry and quadratic deformations of the space-time conformal superalgebra, Journal of physics A: Mathematical and theoretical 51, 14, 145203, 1-27

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