×

Unitary highest weight representations of noncompact supergroups. (English) Zbl 0655.17009

The oscillator method for the construction of unitary irreducible representations (UIRs) of noncompact groups [the author and C. Saclioglu, Commun. Math. Phys. 87, 159-179 (1982; Zbl 0498.22019)] has been extended to the case of noncompact groups [I. Bars and the author, ibid. 91, 31-51 (1983; Zbl 0531.17002)]. Making use of the Jordan, 3-graded, decomposition of a supergroup G with respect to its maximal compact subsupergroup K, the oscillator method may be used to construct all the UIRs of lowest weight of SU(n,m), Sp(2n,\({\mathbb{R}})\) and SO *(2n). Here the Jordan decomposition \(L=L_{-1}+L_ 0+L_ 1\) of the Lie algebra of G is replaced by the Kantor, 5-graded, decomposition \(L=L_{-1}+L_{-1/2}+L_ 0+L_{1/2}+L_ 1\) in order to extend the construction procedure to such supergroups as \(OSp(2n+1/2m,{\mathbb{R}}).\) The methods used are appropriate to the case of all simple supergroups whose maximal even subgroups are in the form of a direct product of a compact group with a simple noncompact group. The generalised supercoherent states associated with the UIRs are also defined.
Reviewer: R.C.King

MSC:

17B70 Graded Lie (super)algebras
22E70 Applications of Lie groups to the sciences; explicit representations
17A70 Superalgebras
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] DOI: 10.1016/0370-1573(81)90157-5 · doi:10.1016/0370-1573(81)90157-5
[2] DOI: 10.1016/0375-9474(81)90078-6 · doi:10.1016/0375-9474(81)90078-6
[3] DOI: 10.1016/0370-2693(82)91170-4 · doi:10.1016/0370-2693(82)91170-4
[4] DOI: 10.1007/BF01218560 · Zbl 0498.22019 · doi:10.1007/BF01218560
[5] DOI: 10.1007/BF01206048 · Zbl 0531.17002 · doi:10.1007/BF01206048
[6] DOI: 10.1016/0550-3213(85)90129-4 · doi:10.1016/0550-3213(85)90129-4
[7] DOI: 10.1016/0550-3213(86)90342-1 · doi:10.1016/0550-3213(86)90342-1
[8] DOI: 10.1088/0264-9381/2/2/001 · doi:10.1088/0264-9381/2/2/001
[9] DOI: 10.1016/0370-2693(86)91528-5 · doi:10.1016/0370-2693(86)91528-5
[10] DOI: 10.1016/0370-2693(86)91528-5 · doi:10.1016/0370-2693(86)91528-5
[11] DOI: 10.1016/0550-3213(86)90237-3 · doi:10.1016/0550-3213(86)90237-3
[12] DOI: 10.2307/2372481 · Zbl 0070.11602 · doi:10.2307/2372481
[13] DOI: 10.2307/2372481 · Zbl 0070.11602 · doi:10.2307/2372481
[14] Wolf J. A., J. Math. Mech. 13 pp 489– (1964)
[15] DOI: 10.1016/S1385-7258(62)50051-6 · doi:10.1016/S1385-7258(62)50051-6
[16] DOI: 10.1016/S1385-7258(62)50051-6 · doi:10.1016/S1385-7258(62)50051-6
[17] Kantor I. L., Trudy Sem. Vector. Anal. 16 pp 407– (1972)
[18] DOI: 10.1063/1.524309 · Zbl 0412.17004 · doi:10.1063/1.524309
[19] Günaydin M., Ann. Israel Phys. Soc. 3 pp 279– (1980)
[20] DOI: 10.1016/0001-8708(77)90017-2 · Zbl 0366.17012 · doi:10.1016/0001-8708(77)90017-2
[21] DOI: 10.1063/1.525508 · Zbl 0488.22040 · doi:10.1063/1.525508
[22] DOI: 10.1063/1.525508 · Zbl 0488.22040 · doi:10.1063/1.525508
[23] DOI: 10.1063/1.525508 · Zbl 0488.22040 · doi:10.1063/1.525508
[24] DOI: 10.1103/PhysRevD.12.3810 · doi:10.1103/PhysRevD.12.3810
[25] DOI: 10.1103/PhysRevD.12.3810 · doi:10.1103/PhysRevD.12.3810
[26] DOI: 10.1103/PhysRevD.12.3810 · doi:10.1103/PhysRevD.12.3810
[27] DOI: 10.1103/PhysRevD.12.3810 · doi:10.1103/PhysRevD.12.3810
[28] DOI: 10.1016/0550-3213(84)90164-0 · Zbl 1223.81116 · doi:10.1016/0550-3213(84)90164-0
[29] DOI: 10.1007/BF01942332 · Zbl 0452.22020 · doi:10.1007/BF01942332
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.