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Representations of Lie superalgebras. I: Extensions of representations of the even part. (English) Zbl 0674.17010
In this paper, representations of Lie superalgebras are studied. Since the classification of all finite-dimensional irreducible representations of basic classical Lie superalgebras by means of their highest weight [V. G. Kac, Adv. Math. 26, 8-96 (1977; Zbl 0366.17012)], and the analysis of typical representations [V. G. Kac, Lect. Notes Math. 676, 597-626 (1978; Zbl 0388.17002)], most effort has gone in analyzing the remaining atypical highest weight representations. Here, a completely different approach is taken.
The authors discuss the following problems, for $${\mathfrak g}={\mathfrak g}_ 0\oplus {\mathfrak g}_ 1$$ any Lie superalgebra: Problem 1 (resp. 1bis): Given an irreducible (resp. a reducible) representation $$\rho$$ of $${\mathfrak g}_ 0$$ on $$V_ 0$$, do there exist irreducible representations ($$\pi$$,V) of $${\mathfrak g}$$ extending $$(\rho,V_ 0)?$$ Problem 2 (resp. 2bis): Given an irreducible (resp. a reducible) unitary representation $$(\rho,V_ 0)$$ of $${\mathfrak g}_ 0$$, do there exist irreducible unitary extensions ($$\pi$$,V)?
Having these problems in mind, a theory is developed the main result of which is that such extensions are possible provided a bilinear map B: $${\mathfrak g}\times {\mathfrak g}_ 1\to {\mathfrak gl}(V_ 0)$$ exists satisfying certain conditions. Although the mathematics involved in this theory may be interesting, the applicability turns out to be rather disappointing. This is due to the fact that the restriction of an irreducible $${\mathfrak g}$$-module to $${\mathfrak g}_ 0$$ is in general reducible (and for the converse problem it turns out to be very difficult to see which reducible $${\mathfrak g}_ 0$$-modules can be extended to an irreducible $${\mathfrak g}$$- module), and that a finite-dimensional irreducible $${\mathfrak g}$$-module is in general not unitary.
Finally, some examples are discussed. The finite example is $${\mathfrak osp}(1,2)$$ [see also J. W. B. Hughes, J. Math. Phys. 22, 245-250 (1981; Zbl 0456.22013)]; the second example is sl(2,1), where only problem 2 (not 2bis) is solved completely [see also M. Marcu, J. Math. Phys. 21, 1277-1283 (1980; Zbl 0449.17001)] for finite-dimensional representations, and J. B. W. Hughes, Ann. Isr. Phys. Soc. 3, 320- 322 (1980; Zbl 0674.17009) for infinite-dimensional representations].