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Oscillator representations for orthosymplectic algebras. (English) Zbl 0688.17002
An \(\epsilon\)-graded Lie algebra (\(\epsilon\) GLA) is defined as a \(\Gamma\)-graded algebra with a commutation factor \(\epsilon\) such that the “\(\epsilon\) skew symmetry” and “\(\epsilon\)-Jacobi identity” are satisfied. When \(\epsilon (\alpha,\beta)=1\), the definition reduces to that of an ordinary graded Lie algebra; when \(\Gamma ={\mathbb{Z}}_ 2\) and \(\epsilon (\alpha,\beta)=(-1)^{\alpha \beta}\) for \(\alpha,\beta \in {\mathbb{Z}}_ 2\), it reduces to the definition of a Lie superalgebra [V. G. Kac, Adv. Math. 26, 8-96 (1977; Zbl 0366.17012)]. One of the series of simple Lie superalgebras is the orthosymplectic series. In classical Lie algebra theory, the spin representation of the orthogonal Lie algebra and the metaplectic representation of the symplectic Lie algebra are of great importance. Here, a similar construction unifies these representations and leads to the oscillator representation (or the singleton representation) of the orthosymplectic Lie superalgebra \({\mathfrak osp}(p,2q).\)
An \(\epsilon\)-Heisenberg algebra is introduced by extending the usual definition. Then a Clifford-Weyl algebra is defined for the case of \(\epsilon\) GLA’s; if \(\epsilon\) is symmetric it includes a Clifford algebra and if \(\epsilon\) is skew symmetric it includes a Weyl algebra. Representations of the Clifford-Weyl algebra are obtained from those of the \(\epsilon\)-Heisenberg algebra. Then the orthosymplectic Lie superalgebra is shown to be realizable in the Clifford-Weyl algebra, thus proving that representations of \({\mathfrak osp}(p,2q)\) can be obtained from those of the \(\epsilon\)-Heisenberg algebra. Unitarity and super unitarity of the representations are investigated.
Reviewer: J.Van der Jeugt

17A70 Superalgebras
17B70 Graded Lie (super)algebras
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[1] Atiyah, M.F; Bott, R; Shapiro, A, Clifford modules, Topology, 3, Suppl. 1, 3-38, (1964) · Zbl 0146.19001
[2] Barbasch, D; Vogan, D, Primitive ideals and orbital integrals in complex classical groups, Math. ann., 259, 153-199, (1982) · Zbl 0489.22010
[3] Berezin, F.A, Introduction to superanalysis, (1987), Reidel Dordrecht
[4] Bernstein, I.N, The analytic continuation of generalized functions with respect to a parameter, Functional anal. appl., 6, 26-40, (1972)
[5] Bohm, A; Kmiecik, M; Boya, L.J, Representation theory of super-conformal quantum mechanics, J. math. phys., 29, 1163-1170, (1988) · Zbl 0654.17013
[6] Bourbaki, N, Algèbre, (1959), Hermann Paris, Chap. 9 · Zbl 0102.25503
[7] Corwin, L; Ne’eman, Y; Sternberg, S, Graded Lie algebras in mathematics and physics (Bose-Fermi symmetry), Rev. modern phys., 47, 573-603, (1975) · Zbl 0557.17004
[8] Duistermaat, J, Fourier integral operators, (1973), Courant Institute of Math. Soc New York · Zbl 0272.47028
[9] Furutsu, H, On representations of Lie superalgebras, II, (), 147-150 · Zbl 0674.17011
[10] Furutsu, H, Representations of Lie superalgebras. II. unitary representations of Lie superalgebras of type A(n, 0), (1988), preprint · Zbl 0701.17013
[11] Furutsu, H; Hirai, T, Representations of Lie superalgebras. I. extensions of representations of the even part, J. math. Kyoto univ., 28, 695-749, (1988) · Zbl 0674.17010
[12] Günaydin, M, Unitary highest weight representations of non-compact supergroups, J. math. phys., 29, 1275-1282, (1988) · Zbl 0655.17009
[13] Hölmander, L, The analysis of linear partial differential operators I, (1983), Springer-Verlag New York/Berlin
[14] Howe, R, Remarks on classical invariant theory, Trans. amer. math. soc., 313, 539-570, (1989) · Zbl 0674.15021
[15] Howe, R, On the role of Heisenberg group in harmonic analysis, Bull. amer. math. soc., 3, 821-843, (1980) · Zbl 0442.43002
[16] Howe, R, Wave front sets of representations of Lie groups, () · Zbl 0494.22010
[17] Humphreys, J.E, Introduction to Lie algebras and representation theory, (1972), Springer-Verlag New York/Berlin · Zbl 0254.17004
[18] Joseph, A, The minimal orbit in a simple Lie algebra and its associated maximal ideal, Ann. sci. école norm. sup., 9, 1-30, (1976) · Zbl 0346.17008
[19] Kac, V, Lie superalgebras, Adv. in math., 26, 8-96, (1977) · Zbl 0366.17012
[20] Kashiwara, M; Vergne, M, On the Segal-shale-Weil representations and harmonie polynomials, Invent. math., 44, 1-47, (1978) · Zbl 0375.22009
[21] Kashiwara, M; Vergne, M, K-types and singular spectrum, Lecture notes in math., 728, 177-200, (1979)
[22] King, D.R, The character polynomial of the annihilator of an irreducible harish chandra module, Amer. J. math., 103, 1195-1240, (1981) · Zbl 0486.17003
[23] Quint, S.R, Representations of solvable Lie-groups, () · Zbl 0347.22002
[24] Sattinger, D.H; Weaver, O.L, Lie groups and algebras with applications to physics, geometry, and mechanics, (1986), Springer-Verlag New York/Berlin · Zbl 0595.22017
[25] Scheunert, M, Graded tensor calculus, (1982), Universität Bonn, Phisikalische Institute, preprint · Zbl 0547.17003
[26] Shale, D, Linear symmetries of free boson fields, Trans. amer. math. soc., 103, 149-167, (1962) · Zbl 0171.46901
[27] Sternberg, S; Wolf, J.A, Hermitian Lie algebras and metaplectic représentations, I, Trans. amer. math. soc., 238, 1-43, (1978) · Zbl 0386.22010
[28] Torasso, P, Sur le caractère de la représentation de shale-Weil de mn(n, \(R\)) et sp(n, \(C\)), Math. ann., 252, 53-86, (1980) · Zbl 0452.22015
[29] Vogan, D, Gelfand-Kirillov dimension for harish-chandra modules, Invent. math., 48, 75-98, (1978) · Zbl 0389.17002
[30] Weil, A, Sur certains groupes d’opérateurs unitaires, Acta math., 111, 143-211, (1964) · Zbl 0203.03305
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