We give a construction of the Lie algebras of the non-compact groups appearing in four dimensional supergravity theories in terms of boson operators. Our construction parallels very closely their emergence in supergravity and is an extension of the well-known construction of the Lie algebras of the non-compact groups SP(2n, R and SO(2n)* from boson operators transforming like a fundamental representation of their maximal compact subgroup U(n). However this extension is non-trivial only for n≥4 and stops at n = 8 leading to the Lei algebras of SU(4) × SU(1, 1), SU(1, 1), SU(5, 1), SO(12)* and E7(7). We then give a general construction of an infinite class of unitary irreducible representations of the respective non-compact groups (except for E7(7) and SO(12)* obtained from the extended construction). We illustrate our construction with the examples of SU(5, 1) and SO(12)*.
All Science Journal Classification (ASJC) codes
- Nuclear and High Energy Physics