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We construct a family of non-weight modules which are free $U (𝔥)$ -modules of rank 2 over the $N = 1$ super Schrödinger algebra in $(1 + 1)$ -dimensional spacetime. We determine the isomorphism classes of these modules. In particular, free $U (𝔥)$ -modules of rank 2 over $𝔬𝔰𝔭 (1 | 2)$ are also constructed and classified. Moreover, we obtain the sufficient and necessary conditions for such modules to be simple.

Normalisation d'une représentation non linéaire d'une algèbre de Lie

Didier Arnal, Mabrouk Benammar, Mohamed Selmi (1988)

Annales de la Faculté des sciences de Toulouse : Mathématiques

On a Construction of Representations and a Problem of Enright.

Vinay V. Deodhar (1980)

Inventiones mathematicae

On a theorem of E. Cartan

B. Морозов (1943)

Matematiceskij sbornik

On affine Kac-Moody Lie algebras

Thomas N. Vougiouklis (1985)

Commentationes Mathematicae Universitatis Carolinae

On bounded generalized Harish-Chandra modules

Ivan Penkov, Vera Serganova (2012)

Annales de l’institut Fourier

Let $𝔤$ be a complex reductive Lie algebra and $𝔨 \subset 𝔤$ be any reductive in $𝔤$ subalgebra. We call a $(𝔤, 𝔨)$ -module $M$ bounded if the $𝔨$ -multiplicities of $M$ are uniformly bounded. In this paper we initiate a general study of simple bounded $(𝔤, 𝔨)$ -modules. We prove a strong necessary condition for a subalgebra $𝔨$ to be bounded (Corollary 4.6), i.e. to admit an infinite-dimensional simple bounded $(𝔤, 𝔨)$ -module, and then establish a sufficient condition for a subalgebra $𝔨$ to be bounded (Theorem 5.1). As a result we are able to...

On category $𝒪$ for cyclotomic rational Cherednik algebras

Iain G. Gordon, Ivan Losev (2014)

Journal of the European Mathematical Society

We study equivalences for category $𝒪_{p}$ of the rational Cherednik algebras $𝐇_{p}$ of type $G_{ℓ} (n) = {(μ_{ℓ})}^{n} ⋊ 𝔖_{n}$ : a highest weight equivalence between $𝒪_{p}$ and $𝒪_{σ (p)}$ for $σ \in 𝔖_{ℓ}$ and an action of $𝔖_{ℓ}$ on an explicit non-empty Zariski open set of parameters $p$ ; a derived equivalence between $𝒪_{p}$ and $𝒪_{p^{'}}$ whenever $p$ and $p^{'}$ have integral difference; a highest weight equivalence between $𝒪_{p}$ and a parabolic category $𝒪$ for the general linear group, under a non-rationality assumption on the parameter $p$ . As a consequence, we confirm special cases of conjectures...

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Lie algebras

Malcev-Moduln.

Matrici simmetriche quantiche e teoria delle rappresentazioni

Matrix canonical realizations of the Lie algebra $o (m, n)$ . I. Basic formulæ and classification

Minimal realizations of classical simple Lie algebras through deformations

Modules de plus haut poids unitarisables sur la super-algèbre de Virasoro $N = 2$ tordue

$𝔫$ -coinvariants des $𝔤$ -modules $𝔫$ -localement nilpotents

New Unitary Representations of Loop Groups.

Nilpotent cohomology of the oscillator representation.

Noncomplete affine structures on Lie algebras of maximal class.

Non-weight modules over the super Schrödinger algebra