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Indecomposable representations with invariant inner product

A theory of the Gupta-Bleuler triplet

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Abstract

Consequences of the existence of an invariant (necessarily indefinite) non-degenerate inner product for an indecomposable representation π of a groupG on a space\(\mathfrak{H}\) are studied. If π has an irreducible subrepresentation π1 on a subspace\(\mathfrak{H}_1 \), it is shown that there exists an invariant subspace\(\mathfrak{H}_2 \) of\(\mathfrak{H}\) containing\(\mathfrak{H}_1 \) and satisfying the following conditions: (1) the representation π #1 =π mod\(\mathfrak{H}_2 \) on\(\mathfrak{H}\) mod\(\mathfrak{H}_2 \) is conjugate to the representation (π1,\(\mathfrak{H}_1 \)), (2)\(\mathfrak{H}_1 \) is a null space for the inner product, and (3) the induced inner product on\(\mathfrak{H}_2 \) mod\(\mathfrak{H}_1 \) is non-degenerate and invariant for the representation

$$\pi _2 = (\pi _2 |_{\mathfrak{H}_2 } )\bmod \mathfrak{H}_1 ,$$

a special example being the Gupta-Bleuler triplet for the one-particle space of the free classical electromagnetic field with\(\mathfrak{H}_1 \)=space of longitudinal photons and\(\mathfrak{H}_2 \)=the space defined by the subsidiary condition.

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Communicated by H. Araki

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Araki, H. Indecomposable representations with invariant inner product. Commun.Math. Phys. 97, 149–159 (1985). https://doi.org/10.1007/BF01206183

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