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We define and study a multiplicity free covering of a graded manifold. As an application of our research we give a new conceptual proof of the theorem about equivalence of categories of graded manifolds and symmetric $n$-fold vector bundles., Comment: 17 pages

We introduce and investigate the notion of a $\mathbb Z$-graded covering for a supermanifold. More precisely, Donagi and Witten suggested a construction of the first obstruction class for splitting of a supermanifold via differential operators. We prove that an infinite prolongation of this construction satisfies some universal properties and can be seen as a covering of a supermanifold in the category of graded manifolds., Comment: The article is a continuation of the unfinished paper arXiv:2103.00665

Any complex-analytic vector bundle $\mathbb E$ admits naturally defined homotheties $\phi_{\alpha}$, $\alpha\in \mathbb C^*$, i.e. $\phi_{\alpha}$ is the multiplication of a local section by a complex number $\alpha$. We investigate the question when such automorphisms can be lifted to a non-split supermanifold corresponding to $\mathbb E$. Further, we compute the automorphism supergroup of a $\Pi$-symmetric super-Grassmannian $\Pi\!\operatorname{Gr}_{n,k}$, and, using Galois cohomology, we classify the real structures on $\Pi\!\operatorname{Gr}_{n,k}$ and compute the corresponding supermanifolds of real points., Comment: 42 Pages, appendix written by Mikhail Borovoi. A missed case fixed, the paper has been significantly revised

We generalize the Donagi and Witten construction of a first obstruction class for splitting of a supermanifold via differential operators using the theory of $n$-fold vector bundles and graded manifolds. Applying the generalized Donagi--Witten construction we obtain a family of embeddings of the category of supermanifolds into the category of $n$-fold vector bundles and into the category of graded manifolds. This leads to a realization of any non-split supermanifold in terms of a collection of vector bundles and some morphism between them. Further we study the images of these embeddings into the category of graded manifolds in the case of a Lie supergroup and a Lie superalgebra. We show that these images satisfy universal property of a graded covering or a graded semicovering., Comment: 41 pages

We prove that under certain assumptions a supermanifold of flags is rigid, this is its complex structure does not admit any non-trivial small deformation. Moreover under the same assumptions we show that a supermanifold of flags is unique non-split supermanifold with given retract.

We construct a new class of algebras resembling enveloping algebras and generalizing orthogonal Gelfand-Zeitlin algebras and rational Galois algebras studied by [EMV,FuZ,RZ,Har]. The algebras are defined via a geometric realization in terms of sheaves of functions invariant under an action of a finite group. A natural class of modules over these algebra can be constructed via a similar geometric realization. In the special case of a local reflection group, these modules are shown to have an explicit basis, generalizing similar results for orthogonal Gelfand-Zeitlin algebras from [EMV] and for rational Galois algebras from [FuZ]. We also construct a family of canonical simple Harish-Chandra modules and give sufficient conditions for simplicity of some modules.

We compute the Lie superalgebras of holomorphic vector fields on isotropic flag supermanifolds of maximal type corresponding to the Lie superalgebra $\mathfrak{osp}_{2m-1|2n}(\mathbb C)$., Comment: 15 pages

We prove that every orthogonal Gelfand-Zeitlin algebra $U$ acts on its Gelfand-Zeitlin subalgebra $\Gamma$. Considering the dual module, we show that every Gelfand-Zeitlin character of $\Gamma$ is realizable in a $U$-module. We observe that the Gelfand-Zeitlin formulae can be rewritten using divided difference operators. It turns out that the action of the latter operators on $\Gamma$ gives rise to an explicit basis in a certain Gelfand-Zeitlin submodule of the dual module mentioned above. This gives, generically, both in the case of regular and singular Gelfand-Zeitlin characters, an explicit construction of simple modules which realize given Gelfand-Zeitlin characters., Comment: v3: some corrections following the referee reports

We give an elementary construction of a $p\geq 1$-singular Gelfand-Tsetlin $\mathfrak{gl}_n(\mathbb C)$-module in terms of local distributions. This is a generalization of the universal $1$-singular Gelfand-Tsetlin $\mathfrak{gl}_n(\mathbb C)$-module obtained in [FGR1]. We expect that the family of new Gelfand-Tsetlin modules that we obtained will lead to a classification of all irreducible $p>1$-singular Gelfand-Tsetlin modules. So far such a classification is known only for singularity $n=1$., Comment: 11 pages

This paper is devoted to an elementary new construction of $1$-singular Gelfand-Tsetlin modules using complex geometry. We introduce a universal ring $\mathcal D_o$ together with the vector space $\mathcal S=\mathcal S(\mathcal D_o)$ with basis $\mathcal B_o = \mathcal B(\mathcal D_o)$ formed from some local distributions such that $\mathcal S$ is a natural $\mathcal D_o$-module. For any homomorphism of rings $\mathcal U(\mathfrak{h}) \to \mathcal D_o$, where $\mathfrak{h}$ is a Lie algebra, it follows that $\mathcal S$ is also an $\mathfrak{h}$-module. We observe that the homomorphism of rings constructed in [FO] is a homomorphism of type $\mathcal U(\mathfrak{gl}_n(\mathbb C)) \to \mathcal D_o$. Using this observation we obtain a construction of the universal $1$-singular Gelfand-Tsetlin $\mathfrak{gl}_n(\mathbb C)$-module from [FRG]., Comment: 7 pages. The version accepted for publication in Differential Geometry and its Applications

The method of double extension, introduced by A.~Medina and Ph.~Revoy, is a procedure which decomposes a Lie algebra with an invariant symmetric form into elementary pieces. Such decompositions were developed for other algebras, for instance for Lie superalgebras and associative algebras, Filippov $n$-algebras and Jordan algebras. The aim of this note is to find a unified approach to such decompositions using the derived bracket formalism. More precisely, we show that any commutative $n$-ary superalgebra with a skew-symmetric invariant form can be obtained inductively by taking orthogonal sums and generalized double extensions., Comment: 9 pages

Vector bundles and double vector bundles, or $2$-fold vector bundles, arise naturally for instance as base spaces for algebraic structures such as Lie algebroids, Courant algebroids and double Lie algebroids. It is known that all these structures possess a unified description using the language of super\-geometry and graded manifolds of degree $\leq 2$. Indeed, a link has been established between the super and classical pictures by the geometrization process, leading to an equivalence of the category of graded manifolds of degree $\leq 2$ and the category of (double) vector bundles with additional structures. In this paper we study the geometrization process in the case of $\mathbb Z^r$-graded manifolds of type $\Delta$, where $\Delta$ is a certain weight system and $r$ is the rank of $\Delta$. We establish an equivalence between a subcategory of the category of $n$-fold vector bundles and the category of graded manifolds of type $\Delta$., Comment: 56 pages, this version was accepted for publication in Letters in Mathematical Physics

These notes are devoted to the multiple generalization of a Lie algebra introduced by A.M.Vinogradov and M.M.Vinogradov. We compare definitions of such algebras in the usual and invariant case. Furthermore, we show that there are no simple $n$-ary Lie algebras of type $(n-1,l)$ for $l>0$.

The main result of this paper is the computation of the Lie superalgebras of holomorphic vector fields on complex flag supermanifolds, introduced by Yu.I.Manin. We prove that with several exceptions any holomorphic vector field is fundamental with respect to the natural action of the Lie superalgebra $\mathfrak {gl}_{m|n}(\mathbb C)$., Comment: arXiv admin note: text overlap with arXiv:1506.02295

We study $n$-ary commutative superalgebras and $L_{\infty}$-algebras that possess a skew-symmetric invariant form, using the derived bracket formalism. This class of superalgebras includes for instance Lie algebras and their $n$-ary generalizations, commutative associative and Jordan algebras with an invariant form. We give a classification of anti-commutative $m$-dimensional $(m-3)$-ary algebras with an invariant form, and a classification of real simple $m$-dimensional Lie $(m-3)$-algebras with a positive definite invariant form up to isometry. Furthermore, we develop the Hodge Theory for $L_{\infty}$-algebras with a symmetric invariant form, and we describe quasi-Frobenius structures on skew-symmetric $n$-ary algebras., Comment: 27 pages