TY - JOUR

T1 - A unified approach to the minimal unitary realizations of noncompact groups and supergroups

AU - Günaydin, Murat

AU - Pavlyk, Oleksandr

PY - 2006/9/1

Y1 - 2006/9/1

N2 - We study the minimal unitary representations of non-compact groups and supergroups obtained by quantization of their geometric realizations as quasi-conformal groups and supergroups. The quasi-conformal groups G leave generalized light-cones defined by a quartic norm invariant and have maximal rank subgroups of the form H × SL(2, ℝ) such that G/H × SL(2, ℝ) are para-quaternionic symmetric spaces. We give a unified formulation of the minimal unitary representations of simple non-compact groups of type A 2, G2, D4, F4, E6, E 7, E8 and Sp(2n, ℝ). The minimal unitary representations of Sp(2n, ℝ) are simply the singleton representations and correspond to a degenerate limit of the unified construction. The minimal unitary representations of the other noncompact groups SU(m, n), SO(m, n), SO*(2n) and SL(m, ℝ) are also given explicitly. We extend our formalism to define and construct the corresponding minimal representations of non-compact supergroups G whose even subgroups are of the form H × SL(2, ℝ). If H is noncompact then the supergroup G does not admit any unitary representations, in general. The unified construction with H simple or Abelian leads to the minimal representations of G(3), F(4) and O Sp(n|2, ℝ) (in the degenerate limit). The minimal unitary representations of O Sp(n|2, ℝ) with even subgroups SO(n) × SL(2, ℝ) are the singleton representations. We also give the minimal realization of the one parameter family of Lie superalgebras D(2, 1; σ).

AB - We study the minimal unitary representations of non-compact groups and supergroups obtained by quantization of their geometric realizations as quasi-conformal groups and supergroups. The quasi-conformal groups G leave generalized light-cones defined by a quartic norm invariant and have maximal rank subgroups of the form H × SL(2, ℝ) such that G/H × SL(2, ℝ) are para-quaternionic symmetric spaces. We give a unified formulation of the minimal unitary representations of simple non-compact groups of type A 2, G2, D4, F4, E6, E 7, E8 and Sp(2n, ℝ). The minimal unitary representations of Sp(2n, ℝ) are simply the singleton representations and correspond to a degenerate limit of the unified construction. The minimal unitary representations of the other noncompact groups SU(m, n), SO(m, n), SO*(2n) and SL(m, ℝ) are also given explicitly. We extend our formalism to define and construct the corresponding minimal representations of non-compact supergroups G whose even subgroups are of the form H × SL(2, ℝ). If H is noncompact then the supergroup G does not admit any unitary representations, in general. The unified construction with H simple or Abelian leads to the minimal representations of G(3), F(4) and O Sp(n|2, ℝ) (in the degenerate limit). The minimal unitary representations of O Sp(n|2, ℝ) with even subgroups SO(n) × SL(2, ℝ) are the singleton representations. We also give the minimal realization of the one parameter family of Lie superalgebras D(2, 1; σ).

UR - http://www.scopus.com/inward/record.url?scp=33749408459&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=33749408459&partnerID=8YFLogxK

U2 - 10.1088/1126-6708/2006/09/050

DO - 10.1088/1126-6708/2006/09/050

M3 - Article

AN - SCOPUS:33749408459

VL - 2006

JO - Journal of High Energy Physics

JF - Journal of High Energy Physics

SN - 1126-6708

IS - 9

M1 - 050

ER -