On structure and TKK algebras for Jordan superalgebras

Barbier, Sigiswald; Coulembier, Kevin

doi:10.1080/00927872.2017.1327059

Cited by 7 publications

(6 citation statements)

References 19 publications

(24 reference statements)

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“…With each Jordan (super)algebra one can associate a 3-graded Lie (super)algebra via the TKKconstruction. There exist different TKK-constructions in the literature, see [5] for an overview, but for the spin factor Jordan superalgebra J K all constructions lead to the orthosymplectic Lie superalgebras osp K (m, 2|2n). We will quickly review the Koecher construction.…”

Section: The Tkk Algebramentioning

confidence: 99%

A Fock Model and the Segal-Bargmann Transform for the Minimal Representation of the Orthosymplectic Lie Superalgebra osp(m,2|2n)

Barbier

Claerebout

Bie

2020

SIGMA

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The minimal representation of a semisimple Lie group is a 'small' infinite-dimensional irreducible unitary representation. It is thought to correspond to the minimal nilpotent coadjoint orbit in Kirillov's orbit philosophy. The Segal-Bargmann transform is an intertwining integral transformation between two different models of the minimal representation for Hermitian Lie groups of tube type. In this paper we construct a Fock model for the minimal representation of the orthosymplectic Lie superalgebra osp(m, 2|2n). We also construct an integral transform which intertwines the Schrödinger model for the minimal representation of the orthosymplectic Lie superalgebra osp(m, 2|2n) with this new Fock model.

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Section: The Tkk Algebramentioning

confidence: 99%

A Fock Model and the Segal-Bargmann Transform for the Minimal Representation of the Orthosymplectic Lie Superalgebra osp(m,2|2n)

Barbier

Claerebout

Bie

2020

SIGMA

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“…With each Jordan (super)algebra one can associate a 3-graded Lie (super)algebra via the TKK-construction. There exist different TKK-constructions in the literature, see [29] for an overview. For D α with α = −1 all constructions lead to D(2, 1; α), while for D −1 we either get psl(2|2) or D(2, 1; −1).…”

Section: 3mentioning

confidence: 99%

A Schrödinger model, Fock model and intertwining Segal-Bargmann transform for the exceptional Lie superalgebra $D(2,1;α)$

Barbier¹,

Claerebout²

2021

Preprint

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We construct two infinite-dimensional irreducible representations for D(2, 1; α): a Schrödinger model and a Fock model. Further, we also introduce an intertwining isomorphism. These representations are similar to the minimal representations constructed for the orthosymplectic Lie supergroup and for Hermitian Lie groups of tube type. The intertwining isomorphism is the analogue of the Segal-Bargmann transform for the orthosymplectic Lie supergroup and for Hermitian Lie groups of tube type. Contents 1. Introduction 1.1. Contents 1.2. Notations 2. The Lie superalgebra D(2, 1; α) 2.1. The construction of D(2, 1; α) 2.2. Roots of D(2, 1; α) 2.3. Three grading 2.4. Real forms 3. The Jordan superalgebra D α 3.1. Definition 3.2. The structure algebra str(D α ) 3.3. The TKK-construction for D α 4. Two polynomial realisations 4.1. The Bessel operator 4.2. A polynomial realisation 4.3. The Fock representation 4.4. The Schrödinger representation 5. The Fock space and Bessel-Fischer product 5.1. The Bessel-Fischer product 5.2. Reproducing kernel 6. Properties of the Fock Representation 6.1. Skew-symmetric 6.2. The (g, k)-module F λ 6.3. The Gelfand-Kirillov dimension

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“…With each Jordan (super)algebra one can associated a 3-graded Lie (super)algebra via the TKK-construction. There exists different TKK-constructions in the literature, see [22] for an overview, but for the spin factor Jordan superalgebra J K all constructions lead to the orthosymplectic Lie superalgebras osp K (m, 2|2n). We will quickly review the Koecher construction.…”

Section: Preliminaries and Notationsmentioning

confidence: 99%

A Fock Model and the Segal-Bargmann Transform for the Minimal Representation of the Orthosymplectic Lie Superalgebra $\mathfrak{osp}(m,2|2n)$

Barbier,

Claerebout,

De Bie

2020

Preprint

Self Cite

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The minimal representation of a semisimple Lie group is a 'small' infinite-dimensional irreducible unitary representation. It is thought to correspond to the minimal nilpotent coadjoint orbit in Kirillov's orbit philosophy. The Segal-Bargmann transform is an intertwining integral transformation between two different models of the minimal representation for Hermitian Lie groups of tube type.In this paper we construct a Fock model for the minimal representation of the orthosymplectic Lie superalgebra osp(m, 2|2n). We also construct an integral transform which intertwines the Schrödinger model for the minimal representation of the orthosymplectic Lie superalgebra osp(m, 2|2n) with this new Fock model.

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On structure and TKK algebras for Jordan superalgebras

Cited by 7 publications

References 19 publications

A Fock Model and the Segal-Bargmann Transform for the Minimal Representation of the Orthosymplectic Lie Superalgebra osp(m,2|2n)

A Fock Model and the Segal-Bargmann Transform for the Minimal Representation of the Orthosymplectic Lie Superalgebra osp(m,2|2n)

A Schrödinger model, Fock model and intertwining Segal-Bargmann transform for the exceptional Lie superalgebra $D(2,1;α)$

A Fock Model and the Segal-Bargmann Transform for the Minimal Representation of the Orthosymplectic Lie Superalgebra $\mathfrak{osp}(m,2|2n)$

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