Volume 151B, number 5,6 PHYSICS LETTERS 21 February 1985 ALL POSSIBLE DE SITTER SUPERALGEBRAS AND THE PRESENCE OF GHOSTS Jerzy LUKIERSKI and Anatol NOWICKI 1 Physikalisches Institut, Universitiit Bonn, Nussallee 12, D-5300 Bonn 1, Fed. Rep. Germany Received 1 October 1984 Revised manuscript received 30 November 1984 De Sitter superalgebras which supersymmetrize SO(d; 1) by introducing spinorial supercharges, exist for d = 2, 3, 4 and 5. It is shown however that it is possible only for d = 2 to write a nontrivial representation of a de Sitter superalgebra in Hilbert space, with positive-definite metric. 1. We consider the de Sitter superalgebras which satisfy the following two conditions: (i) the bosonic sector should have the form SO(d, 1) G, where G describes some Lie algebra of internal sector; (ii) we assume that the fermionic supercharges transform under SO(d, 1) as its fundamental spinorial representations. From the condition (ii) it follows that de Sitter superalgebras exist only for those d for which the spinorial covering of SO(d, 1) is a classical Lie group. It is easy to check that such a covering, denoted by SO(d, 1) exists only for d = 2,3,4, 5 (see e.g. ref. [1 ]) and we obtain SO(2, 1) = Sp(2; R), SO(3, 1) = SL(2; C)- Sp(2;C), SO(4, 1) = U(1, l; H) - USp(2, 2; C), SO(5, 1) = SL(2; H) =- SU*(4). Recently [2-5] explicit formulae were shown su- persymmetrizing d = 4 de Sitter algebra 0(4, 1) -~ U(1, 1 ;H) which form the quaternionic superalgebra UU~(1,1 ;N; H), with the bosonic sector U(1,1 ; H) X Us(N; H) = 0(4, 1) O*(2N), with O*(2N) describing the internal symmetry sector. The basic relation of UUa(1, 1 ;N; H) in complex notation can be written 1 Institute of Teachers Training and Educational Research, ul. Davida la, 50-527 Wroclaw, Poland. as follows (see e.g. ref. [3]): (QiA , QJB) = 6i](~'ab rO)ABMab + ½ (rO)AB( t4 + itS), * -' , ,QI~ )_ ~ i(P5)AB(tg (1) where {Pa, I'b} = 2~?ab [rlab = diag(-1, 1, 1, 1, 1); a, b = 0, 1, 2, 3, 5], ~ab = ~[ra, rb],Mab = -Mbq. are 10 generators of O(4, 1), and t//n = -tjim, ti~ = t12~ (m = 1, 3, 4; i,j = 1 ... A r) span N(2N - 1) generators of the internal sector O*(2N) -= O(N, H) with the fol- lowing algebra: [t~ j,tkr 1]=6ik(EA)rstil A -- 6il(EA)srt ~" + Pru (~]l(EA)ustiA k- 6ik(EA )sutl/A), (2) [t~', 41] = 6 i l t ~ - 6/k d + gABPBC(6fltick--$iktlic), where gAB = diag(-1, -1,-1, I), PAB=diag(--1,1,--1,1) r,s,u=l,2,3, A,B,C=I...4, i,],k,l=l...N. and E A denotes the real 4 × 4 matrix realization of the quaternionic algebra E A - (el, 1), where eie ] = -6ii + eqkek" For N = 1 (simple the de Sitter supersym. metry) we obtain {QA , QB} = (~ab I'O)AB Mab , {QA, Q~) = --(['5)AB T, (3) where T = T211 describes the abelian 0(2) charge. 382