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**Unitary representations of noncompact supergroups.**
*(English)*
Zbl 0531.17002

Lie superalgebras have been studied as a model of the symmetries of elementary particles [e.g., J. Wess and B. Zumino, Supergauge transformations in four dimensions, Nucl. Phys. B 70, 39–50 (1974)], and as interesting mathematical objects in their own right [e.g. V. G. Kac, Adv. Math. 26, 8–96 (1977; Zbl 0366.17012)]. In this paper, an investigation is made of certain representations of certain Lie superalgebras. After an introduction and survey of the literature, the authors show how several classical series of Lie superalgebras can be constructed out of bilinear combinations of raising and lowering operators for a collection of bosons and fermions. The example known as \(\mathrm{SU}(m,p/n+q)\) is emphasized; it is treated by analogy to the classical Lie algebra \(\mathfrak{su}(m,p)\). In particular, it has a “compact” subalgebra \(\mathrm{S}(\mathrm{U}(m/n)\times \mathrm{U}(p/q)),\) analogous to \(\mathfrak{s}(\mathfrak{u}(m)\times \mathfrak{u}(p)),\) and lowering and raising subalgebras analogous to nilpotent subalgebras of the complexification of \(\mathfrak{su}(m,p)\).

The authors then attempt to construct “unitary” irreducible representations in the Fock space of several copies of the collection of bosons and fermions referred to above. The construction proceeds from a representation of the ”compact” subalgebra on a subspace which is annihilated by the ”lowering” subalgebra. Unfortunately, it is not clear that many such representations exist. Also, it needs to be proved that the formal exponentiation of unbounded operators, which the authors indulge in, can be rigorously defined. Lacking this degree of rigour, it remains unproved that “unitary” representations of \(\mathrm{SU}(m,p/n+q)\) have been defined.

The next section attempts a construction analogous to the holomorphic discrete series of representations of \(\mathrm{SU}(m,p)\). In the Fock space, the authors define a family of “coherent” states, whose transformation properties are analogous to those of points in the Hermitian symmetric space of \(\mathrm{SU}(m,p)\). The authors state that these vectors span the representation space previously constructed. However, this is not proved.

The final section shows how the first construction can be generalized to other Lie superalgebras. An attempt is made to prove that the representations are irreducible, by showing that the Casimir operator, and the higher Casimir invariants, reduce to scalars. This is of course necessary, for the representation to be irreducible, but one cannot agree that it is sufficient.

The authors then attempt to construct “unitary” irreducible representations in the Fock space of several copies of the collection of bosons and fermions referred to above. The construction proceeds from a representation of the ”compact” subalgebra on a subspace which is annihilated by the ”lowering” subalgebra. Unfortunately, it is not clear that many such representations exist. Also, it needs to be proved that the formal exponentiation of unbounded operators, which the authors indulge in, can be rigorously defined. Lacking this degree of rigour, it remains unproved that “unitary” representations of \(\mathrm{SU}(m,p/n+q)\) have been defined.

The next section attempts a construction analogous to the holomorphic discrete series of representations of \(\mathrm{SU}(m,p)\). In the Fock space, the authors define a family of “coherent” states, whose transformation properties are analogous to those of points in the Hermitian symmetric space of \(\mathrm{SU}(m,p)\). The authors state that these vectors span the representation space previously constructed. However, this is not proved.

The final section shows how the first construction can be generalized to other Lie superalgebras. An attempt is made to prove that the representations are irreducible, by showing that the Casimir operator, and the higher Casimir invariants, reduce to scalars. This is of course necessary, for the representation to be irreducible, but one cannot agree that it is sufficient.

Reviewer: Christopher J. Henrich (Tinton Falls)

### MSC:

17A70 | Superalgebras |

17B10 | Representations of Lie algebras and Lie superalgebras, algebraic theory (weights) |

81T60 | Supersymmetric field theories in quantum mechanics |

22E70 | Applications of Lie groups to the sciences; explicit representations |

### Keywords:

non-compact supergroups; super Fock space; unitary irreducible representations; Lie superalgebras### Citations:

Zbl 0366.17012
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\textit{I. Bars} and \textit{M. Günaydin}, Commun. Math. Phys. 91, 31--51 (1983; Zbl 0531.17002)

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### References:

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