Unitarization of a singular representation ofSO(p, q)

Binegar, B.; Zierau, Roger

doi:10.1007/bf02099491

Cited by 48 publications

(83 citation statements)

References 11 publications

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“…However it can be checked that this ideal is not of the form of the corresponding classical Joseph ideal for so(m) in [1,2,12]. This is closely related to Remark 3 and the subsequent Remark 5.…”

Section: Remarkmentioning

confidence: 59%

“…and V c = a g ac V a with g ab = g ab . When considering the orthosymplectic Lie superalgebra osp(m|2n) we will always assume m > 4 and n > 1, as in the classical case for so(m) and sp(2n), see [1,12,20]. The orthosymplectic Lie superalgebra can be defined in several ways as the algebra preserving an inner product or element in the tensor space, we restrict to the subsequent definition.…”

Section: Preliminaries and Conventionsmentioning

confidence: 99%

“…Cartan product of the adjoint representation for g = spo(2n|m) A The submodule of g ⊗ g of traceless tensors if m − 2n ∈ {1, 2} A 1 The submodule of g ⊗ g given by A ∩ g ∧ g…”

Section: List Of Notations Spo(2n|m)mentioning

confidence: 99%

“…In particular it would also be of interest to construct general theories as in [1,2,18,26] for Lie superalgebras. For instance there seems to exist an interesting link with (co)adjoint orbits, see [29].…”

Section: Introductionmentioning

confidence: 99%

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Joseph Ideals and Harmonic Analysis for

Coulembier

Somberg

Souček

2013

International Mathematics Research Notices

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The Joseph ideal in the universal enveloping algebra U(so(m)) is the annihilator ideal of the so(m)-representation on the harmonic functions on R m−2 . The Joseph ideal for sp(2n) is the annihilator ideal of the Segal-Shale-Weil (metaplectic) representation. Both ideals can be constructed in a unified way from a quadratic relation in the tensor algebra ⊗g for g equal to so(m) or sp(2n). In this paper we construct two analogous ideals in ⊗g and U(g) for g the orthosymplectic Lie superalgebra osp(m|2n) = spo(2n|m) and prove that they have unique characterizations that naturally extend the classical case. Then we show that these two ideals are the annihilator ideals of respectively the osp(m|2n)-representation on the spherical harmonics on R m−2|2n and a generalization of the metaplectic representation to spo(2n|m). This proves that these ideals are reasonable candidates to establish the theory of Joseph-like ideals for Lie superalgebras. We also discuss the relation between the Joseph ideal of osp(m|2n) and the algebra of symmetries of the super conformal Laplace operator, regarded as an intertwining operator between principal series representations for osp(m|2n). As a side result we obtain the proof of a conjecture of M. Eastwood about the Cartan product of irreducible representations of semisimple Lie algebras made in [Bull.

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Section: Remarkmentioning

confidence: 59%

Section: Preliminaries and Conventionsmentioning

confidence: 99%

“…Cartan product of the adjoint representation for g = spo(2n|m) A The submodule of g ⊗ g of traceless tensors if m − 2n ∈ {1, 2} A 1 The submodule of g ⊗ g given by A ∩ g ∧ g…”

Section: List Of Notations Spo(2n|m)mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Joseph Ideals and Harmonic Analysis for

Coulembier

Somberg

Souček

2013

International Mathematics Research Notices

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“…Interestingly, HS algebras mentioned in the above paragraph all fall into these coset Lie algebras. Much progress has been made on the subject of minimal representations, see for example [40][41][42][43][44][45][46][47][48][49][50] and references therein. In physics literature, the minimal representations of isometry algebras are explored to a large extent by Gunaydin and collaborators [51][52][53][54][55].…”

Section: Jhep05(2014)103mentioning

confidence: 99%

Notes on higher-spin algebras: minimal representations and structure constants

Joung

Mkrtchyan

2014

J. High Energ. Phys.

148

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The higher-spin (HS) algebras relevant to Vasiliev's equations in various dimensions can be interpreted as the symmetries of the minimal representation of the isometry algebra. After discussing this connection briefly, we generalize this concept to any classical Lie algebra and consider the corresponding HS algebras. For sp 2N and so N , the minimal representations are unique so we get unique HS algebras. For sl N , the minimal representation has one-parameter family, so does the corresponding HS algebra. The so N HS algebra is what underlies the Vasiliev theory while the sl 2 one coincides with the 3D HS algebra hs [λ]. Finally, we derive the explicit expression of the structure constant of these algebras -more precisely, their bilinear and trilinear forms. Several consistency checks are carried out for our results.

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Deformations of Quadratic Algebras, the Joseph Ideal for Classical Lie Algebras, and Special Tensors

Somberg

2008

Symmetries and Overdetermined Systems of Partial Differential Equations

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Unitarization of a singular representation ofSO(p, q)

Cited by 48 publications

References 11 publications

Joseph Ideals and Harmonic Analysis for

Joseph Ideals and Harmonic Analysis for

Notes on higher-spin algebras: minimal representations and structure constants

Deformations of Quadratic Algebras, the Joseph Ideal for Classical Lie Algebras, and Special Tensors

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