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Van der Jeugt, Joris

Irreducible representations of the exceptional Lie superalgebras \(D(2,1;\alpha)\). (English) Zbl 0604.17001

J. Math. Phys. 26, 913-924 (1985).
The shift operator technique is used to give a complete analysis of all finite- and infinite-dimensional irreducible representations of the exceptional Lie superalgebras \(D(2,1;\alpha)\). For all cases, the star or grade star conditions for the algebra are investigated. Among the finite- dimensional representations there are no star and only a few grade star representations, but an infinite class of infinite-dimensional star representations is found. Explicit expressions are given for the “doublet” representation of \(D(2,1;\alpha)\). The one missing label problem \(D(2,1;\alpha)\to\text{su}(2)+\text{su}(2)+\text{su}(2)\) is discussed in detail and solved explicitly.

Cited in 35 Documents

MSC:

17B25 Exceptional (super)algebras
17B10 Representations of Lie algebras and Lie superalgebras, algebraic theory (weights)
81R10 Infinite-dimensional groups and algebras motivated by physics, including Virasoro, Kac-Moody, \(W\)-algebras and other current algebras and their representations
22E70 Applications of Lie groups to the sciences; explicit representations

Keywords:

irreducible representations; exceptional Lie superalgebras; star conditions; missing label problem

Citations:

Zbl 0603.17002
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Full Text: DOI

References:

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