Low Degree Morphisms of E(5, 10)-Generalized Verma Modules

Cantarini, Nicoletta; Caselli, Fabrizio

doi:10.1007/s10468-019-09925-0

Cited by 13 publications

(41 citation statements)

References 11 publications

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“…Next we find that for degrees between 11 and 14 there is only one singular vector, it has degree 11 and defines a morphism from M(C 5 ) to M(C 5 * ), where C 5 is the standard sl 5 -module and C 5 * its dual. After that, using the techniques of [2], we show that in degrees between 6 and 10 the only singular vector has degree 7 and it defines a morphism from M(S 2 C 5 ) to M(S 2 C 5 * ). These are precisely the two morphisms, missing in Fig.…”

Section: Introductionmentioning

confidence: 90%

“…We recall the definition and some properties of (generalized) Verma modules over E (5,10), most of which hold in the generality of arbitrary Z-graded Lie superalgebras (for some detailed proofs see [2]).…”

Section: Generalized Verma Modules and Morphismsmentioning

confidence: 99%

“…We recall that a finite Verma module M(V ) is degenerate if and only if it contains a singular vector [2,Proposition 3.3].…”

Section: Generalized Verma Modules and Morphismsmentioning

confidence: 99%

“…It is shown in [2] that the leading term of a singular vector is non-zero, and therefore a singular vector is uniquely determined by its leading term.…”

Section: Generalized Verma Modules and Morphismsmentioning

confidence: 99%

“…In a more recent paper [13] it was proved that these are all singular vectors of degree 1, and also some singular vectors of degree 2,3,4 and 5 have been constructed. In the subsequent paper [2] it was shown that the singular vectors of degree less than or equal to 3 constructed by Rudakov are all singular vectors of degree less than or equal to 3. Actually, the morphisms of degrees 2, 3 and 5 corresponding to singular vectors constructed in [13] are composition of morphisms of degree 1 and 4, and the morphisms of degree 1 and 4 can be arranged in an infinite number of infinite complexes [13].…”

Section: Introductionmentioning

confidence: 99%

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Classification of Degenerate Verma Modules for E(5, 10)

Cantarini

Caselli

Kač

2021

Commun. Math. Phys.

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Given a Lie superalgebra $${\mathfrak {g}}$$ g with a subalgebra $${\mathfrak {g}}_{\ge 0}$$ g ≥ 0 , and a finite-dimensional irreducible $${\mathfrak {g}}_{\ge 0}$$ g ≥ 0 -module F, the induced $${\mathfrak {g}}$$ g -module $$M(F)={\mathcal {U}}({\mathfrak {g}})\otimes _{{\mathcal {U}}({\mathfrak {g}}_{\ge 0})}F$$ M ( F ) = U ( g ) ⊗ U ( g ≥ 0 ) F is called a finite Verma module. In the present paper we classify the non-irreducible finite Verma modules over the largest exceptional linearly compact Lie superalgebra $${\mathfrak {g}}=E(5,10)$$ g = E ( 5 , 10 ) with the subalgebra $${\mathfrak {g}}_{\ge 0}$$ g ≥ 0 of minimal codimension. This is done via classification of all singular vectors in the modules M(F). Besides known singular vectors of degree 1,2,3,4 and 5, we discover two new singular vectors, of degrees 7 and 11. We show that the corresponding morphisms of finite Verma modules of degree 1,4,7, and 11 can be arranged in an infinite number of bilateral infinite complexes, which may be viewed as “exceptional” de Rham complexes for E(5, 10).

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Section: Introductionmentioning

confidence: 90%

Section: Generalized Verma Modules and Morphismsmentioning

confidence: 99%

“…We recall that a finite Verma module M(V ) is degenerate if and only if it contains a singular vector [2,Proposition 3.3].…”

Section: Generalized Verma Modules and Morphismsmentioning

confidence: 99%

“…It is shown in [2] that the leading term of a singular vector is non-zero, and therefore a singular vector is uniquely determined by its leading term.…”

Section: Generalized Verma Modules and Morphismsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Classification of Degenerate Verma Modules for E(5, 10)

Cantarini

Caselli

Kač

2021

Commun. Math. Phys.

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SL(5) Supersymmetry

Cederwall

2021

Fortschritte der Physik

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We consider supersymmetry in five dimensions, where the fermionic parameters are a 2-form under SL(5). Supermultiplets are investigated using the pure spinor superfield formalism, and are found to be closely related to infinite-dimensional extensions of the supersymmetry algebra: the Borcherds superalgebra ℬ(E 4 ), the tensor hierarchy algebra S(E 4 ) and the exceptional superalgebra E(5, 10). A theorem relating ℬ(E 4 ) and E(5, 10) to all levels is given. Introduction and OverviewSupersymmetry provides an extension of bosonic space-time symmetries with fermionic generators. These are generically spinors under space-time rotations (and may also transform under R-symmetry). In certain situations, supersymmetry generators in non-spinorial modules may be considered. The main example is provided by "twisting", where one considers a fermionic generator which is a singlet under some subalgebra.More generically, one may a priori consider an assignment where supersymmetry generators come in a module S of a spacetime "structure group" G, which we think of as corresponding to the double cover of the Lorentz group together with R-symmetry. A supersymmetry algebra will take the form 1 [Q a , Q b ] = c ab m P m , with some invariant tensor c, and the rest of the brackets vanishing. The only condition is that the symmetric product ∨ 2 S contains the vector representation V.Presently, we will consider one specific such assignment, namely when the structure group is G = SL(5), V = 5 and S = 10. The supersymmetry algebra then is 2 [Q mn , Q pq ] = 2𝜖 mnpqr 𝜕 r ,(1.1

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Computation of the Homology of the Complexes of Finite Verma Modules for ${K}_{4}^{\prime }$

Bagnoli

2022

Algebr Represent Theor

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We compute the homology of the complexes of finite Verma modules over the annihilation superalgebra $\mathcal {A}({K}_{4}^{\prime })$ A ( K 4 ′ ) , associated with the conformal superalgebra ${K}_{4}^{\prime }$ K 4 ′ , obtained in Bagnoli and Caselli (J. Math. Phys. 63, 091701, 2022). We use the computation of the homology in order to provide an explicit realization of all the irreducible quotients of finite Verma modules over $\mathcal {A}({K}_{4}^{\prime })$ A ( K 4 ′ ) .

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Low Degree Morphisms of E(5, 10)-Generalized Verma Modules

Cited by 13 publications

References 11 publications

Classification of Degenerate Verma Modules for E(5, 10)

Classification of Degenerate Verma Modules for E(5, 10)

SL(5) Supersymmetry

Computation of the Homology of the Complexes of Finite Verma Modules for ${K}_{4}^{\prime }$

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