Positive energy representations, holomorphic discrete series and finite-dimensional irreps

Dobrev, V. K.

doi:10.1088/1751-8113/41/42/425206

Cited by 14 publications

(21 citation statements)

References 69 publications

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“…As we mentioned the case su(1, 1) is well known -it was studied 60 years ago in the isomorphic form sl(2, R) by Gelfand et al [19] and by Bargmann [20]. In the current setting it was given in [21]. Here we shall only mention that the multiplets contain two ERS/GVMs (cf.…”

Section: Preliminariesmentioning

confidence: 95%

Multiplet classification for SU(n,n)

Dobrev¹

2014

J. Phys.: Conf. Ser.

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Abstract. In the present paper we review our project of systematic construction of invariant differential operators on the example of the non-compact algebras su(n, n) for n = 2, 3, 4. We give explicitly the main multiplets of indecomposable elementary representations and some reduced multiplets. We give explicitly the minimal representations. Due to the recently established parabolic relations the multiplet classification results are valid also for the algebras sl(2n, R) and when n = 2k for the algebras su * (4k) with suitably chosen maximal parabolic subalgebras.

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Section: Preliminariesmentioning

confidence: 95%

Multiplet classification for SU(n,n)

Dobrev¹

2014

J. Phys.: Conf. Ser.

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“…Following results of [107][108][109], [91] we present the main multiplets (which contain the maximal number of ERs with this parabolic) with the explicit parametrization of the ERs in the multiplets in a simple way (helped by the use of the signature entry c):…”

Section: Main Multipletsmentioning

confidence: 99%

“…Highest/lowest weight GVMs are related to other pairs besides χ + 1 . A detailed analysis of these occurrences is done for the conformal algebra so(3, 2) in [91] and for so(4, 2) in [108], [91].…”

Section: Main Multipletsmentioning

confidence: 99%

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8. Invariant Differential Operators for Noncompact Lie Algebras Parabolically Related to Conformal Lie Algebras

2016

Noncompact Semisimple Lie Algebras and Groups

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In the present paper we continue the project of systematic construction of invariant differential operators for non-compact semisimple Lie groups. Our starting points is the class of algebras, which we call 'conformal Lie algebras' (CLA), which have very similar properties to the conformal algebras of Minkowski space-time, though our aim is to go beyond this class in a natural way. For this we introduce the new notion of parabolic relation between two non-compact semisimple Lie algebras G and G ′ that have the same complexification and possess maximal parabolic subalgebras with the same complexification. Thus, we consider the exceptional algebra E 7(7) which is parabolically related to the CLA E 7(−25) , the parabolic subalgebras including E 6(6) and E 6(−26) . Other interesting examples are the orthogonal algebras so(p, q) all of which are parabolically related to the conformal algebra so(n, 2) with p + q = n + 2, the parabolic subalgebras including the Lorentz subalgebra so(n − 1, 1) and its analogs so(p − 1, q − 1). We consider also E 6(6) and E 6(2) which are parabolically related to the hermitian symmetric case E 6(−14) , the parabolic subalgebras including real forms of sl(6). We also give a formula for the number of representations in the main multiplets valid for CLAs and all algebras that are parabolically related to them. In all considered cases we give the main multiplets of indecomposable elementary representations including the necessary data for all relevant invariant differential operators. In the case of so(p, q) we give also the reduced multiplets. We should stress that the multiplets are given in the most economic way in pairs of shadow fields. Furthermore we should stress that the classification of all invariant differential operators includes as special cases all possible conservation laws and conserved currents, unitary or not.

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“…The reason to use the parameter c instead of d is that the parametrization of the ERs in the multiplets is given in a simple intuitive way (cf. [84], [67]):…”

Section: Preliminariesmentioning

confidence: 99%

Classification of Invariant Differential Operators for Non-Compact Lie Algebras via Parabolic Relations

Dobrev

2014

J. Phys.: Conf. Ser.

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In the present paper we review the progress of the project of classification and construction of invariant differential operators for non-compact semisimple Lie groups. Our starting points is the class of algebras, which we called earlier 'conformal Lie algebras' (CLA), which have very similar properties to the conformal algebras of Minkowski space-time, though our aim is to go beyond this class in a natural way. For this we introduced recently the new notion of parabolic relation between two non-compact semisimple Lie algebras G and G ′ that have the same complexification and possess maximal parabolic subalgebras with the same complexification. Thus, we consider the exceptional algebra E 7(7) which is parabolically related to the CLA E 7(−25) . Other interesting examples are the orthogonal algebras so(p, q) all of which are parabolically related to the conformal algebra so(n, 2) with p + q = n + 2, the parabolic subalgebras including the Lorentz subalgebra so(n − 1, 1) and its analogs so(p − 1, q − 1). Further we consider the algebras sl(2n, R) and for n = 2k the algebras su * (4k) which are parabolically related to the CLA su(n, n). Further we consider the algebras sp(r, r) which are parabolically related to the CLA sp(2r, R). We consider also E 6(6) and E 6(2) which are parabolically related to the hermitian symmetric case E 6(−14) .

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Positive energy representations, holomorphic discrete series and finite-dimensional irreps

Cited by 14 publications

References 69 publications

Multiplet classification for SU(n,n)

Multiplet classification for SU(n,n)

8. Invariant Differential Operators for Noncompact Lie Algebras Parabolically Related to Conformal Lie Algebras

Classification of Invariant Differential Operators for Non-Compact Lie Algebras via Parabolic Relations

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