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We study the classical problem of Kostant for Whittaker modules over Lie algebras and Lie superalgebras. We give a sufficient condition for a positive answer to Kostant’s problem for the standard Whittaker modules over reductive Lie algebras. Under the same condition, the positivity of the answer for simple Whittaker modules is reduced to that for simple highest weight modules. We develop several reduction results to reduce the Kostant’s problem for standard and simple Whittaker modules over a type I Lie superalgebra to that for the corresponding Whittaker modules over the even part of this Lie superalgebra. [ABSTRACT FROM AUTHOR]

Recently the quantum toroidal superalgebra ℰ m | n associated with g l m | n was introduced by L. Bezerra and E. Mukhin, which is not a quantum Kac–Moody algebra. The quantum toroidal superalgebra ℰ m | n exploits infinite sequences of generators and relations of the form, which are called Drinfeld realization. In this paper, we use only finite set of generators and relations to define an associative algebra ℰ m | n ′ and show that there exists an epimorphism from ℰ m | n ′ to the quantum toroidal superalgebra ℰ m | n . In particular, the structure of ℰ m | n ′ enjoys some properties like Drinfeld–Jimbo realization. [ABSTRACT FROM AUTHOR]

Two one-parameter families of positively graded Lie superalgebras generated by two elements and two relations that are narrow in the sense of Zelmanov and Shalev are considered. The first family contains the positive part R+ of the Ramond algebra, while the second one contains the positive part NS+ of the Neveu–Schwarz algebra. The results of the article are super analogues of Benoist's theorem on defining the positive part of the Witt algebra by generators and relations. [ABSTRACT FROM AUTHOR]

We construct a Springer-type resolution of singularities of the odd nilpotent cone of the orthosymplectic Lie superalgebras . [ABSTRACT FROM AUTHOR]

In this paper, all symmetric super-biderivations of some Lie superalgebras are determined. As an application, commutative post-Lie superalgebra structures on these Lie superalgebras are also obtained. [ABSTRACT FROM AUTHOR]

Throughout this paper, we show on one hand, that there are nilpotent and solvable Lie superalgebras with infinitely many local superderivations which are not standard superderivations. On the other hand, we show that every local superderivation is a superderivation on the maximal-dimensional solvable Lie superalgebras with model filiform or model nilpotent nilradical. Moreover, we extend the latter result for Leibniz superalgebras by showing that every local superderivation is a superderivation on the maximal-dimensional solvable Leibniz superalgebras with model filiform or model nilpotent non-Lie nilradical. [ABSTRACT FROM AUTHOR]

Sasakian manifolds are odd-dimensional counterpart to Kähler manifolds. They can be defined as contact manifolds equipped with an invariant Kähler structure on their symplectic cone. The quotient of this cone by the homothety action is a complex manifold called Vaisman. We study harmonic forms and Hodge decomposition on Vaisman and Sasakian manifolds. We construct a Lie superalgebra associated to a Sasakian manifold in the same way as the Kähler supersymmetry algebra is associated to a Kähler manifold. We use this construction to produce a self-contained, coordinate-free proof of the results by Tachibana, Kashiwada and Sato on the decomposition of harmonic forms and cohomology of Sasakian and Vaisman manifolds. In the last section, we compute the supersymmetry algebra of Sasakian manifolds explicitly. [ABSTRACT FROM AUTHOR]

In the framework of the Toledano Laredo and Gautam approach, we consider the structures of tensor categories on the analogues of the -category for representations of the super-Yangian of the special linear Lie superalgebra and the quantum loop superalgebra , and study the relation between them. The basic result is the construction of an isomorphism in the category of Hopf superalgebras between completions of the super-Yangian and quantum loop superalgebra equipped with so-called Drinfeld comultiplications. We formulate the theorem on the equivalence of tensor categories of super-Yangian modules and quantum loop superalgebra modules, which enhances the previous result. We also describe the relation between quasitriangular structures and Abelian difference equations, which are defined by the Abelian parts of the universal -matrices. [ABSTRACT FROM AUTHOR]

As for the orthosymplectic Lie superalgebra osp(2, 2), we construct two isospectral super AKNS hierarchies and a non-isospectral super AKNS hierarchy. Furthermore, we present a generalized non-isospectral super AKNS hierarchy. [ABSTRACT FROM AUTHOR]

In this paper, we study the shape and the reduced shape algebra for the Lie superalgebra sl (m , n) . We define quasistandard Young tableaux, and they give a combinatorial basis for the reduced shape algebra, called the diamond cone. There is a natural bijective map from the set of semistandard tableaux with shape λ to the set of quasistandard tableaux with smaller shape: the diamond cone is compatible with the stratification of the reduced shape algebra. [ABSTRACT FROM AUTHOR]

In this paper, we consider any two ideals I and J of a finite-dimensional Lie superalgera G. Then we define the set Sder J I (G) , which contains the set of all superderivations of G which take J to zero and whose images are contained in I and the set Sder c n (G) , which contains the set of all superderivations δ of G such that δ (g) ∈ [ g , G n ] for all g ∈ G . In this article, we have shown that if (φ , θ) is an n-isoclinism between two finite-dimensional Lie superalgebras G and H, then there exists an isomorphism from Sder Z n (G) G n + 1 (G) to Sder Z n (H) H n + 1 (H) . Also, we prove the necessary and sufficient conditions under which Sder c n (G) is isomorphic to certain special subsuperalgebras of Sder (G) . [ABSTRACT FROM AUTHOR]

In this paper, both canonical and noncanonical polynomial representations of strange classical Lie superalgebra P(n) are investigated. It turns out all these representations are undecomposable, and their composition series are obtained. [ABSTRACT FROM AUTHOR]

We study Hom-Lie superalgebras of Heisenberg type. For 3-dimensional Heisenberg Hom-Lie superalgebras we describe their Hom-Lie super structures, compute the cohomology spaces and characterize their infinitesimal deformations. [ABSTRACT FROM AUTHOR]

We prove that the tensor product of a simple and a finite dimensional sl n -module has finite type socle. This is applied to reduce classification of simple q (n) -supermodules to that of simple sl n -modules. Rough structure of simple q (n) -supermodules, considered as sl n -modules, is described in terms of the combinatorics of category O . [ABSTRACT FROM AUTHOR]

We mainly study the super-biderivations of not-finitely graded Lie superalgebras related to generalized super-Virasoro algebras. In particular, we prove that all super-biderivations of not-finitely graded Lie superalgebras related to generalized super-Virasoro algebras are inner. [ABSTRACT FROM AUTHOR]

Using the approach developed by Gautam and Toledano Laredo, we introduce analogues of the category for representations of the Yangian of a special linear Lie superalgebra and the quantum loop superalgebra . We investigate the relation between them and conjecture that these categories are equivalent. [ABSTRACT FROM AUTHOR]

We construct HNN-extensions of Lie superalgebras and prove that every Lie superalgebra embeds into any of its HNN-extensions. Then as an application we show that any Lie superalgebra with at most countable dimension embeds into a two-generator Lie superalgebra. [ABSTRACT FROM AUTHOR]

In this paper, we study Rota–Baxter operators and super O -operator of associative superalgebras, Lie superalgebras, pre-Lie superalgebras and L-dendriform superalgebras. Then we give some properties of pre-Lie superalgebras constructed from associative superalgebras, Lie superalgebras and L-dendriform superalgebras. Moreover, we provide all Rota–Baxter operators of weight zero on complex pre-Lie superalgebras of dimensions 2 and 3. [ABSTRACT FROM AUTHOR]

Balinsky-Novikov superalgebras were introduced by Balinsky for constructing super-Vira-soro type Lie superalgebras. In this paper, we give sufficient and necessary conditions for a Lie superalgebra generalized by a Balinsky-Novikov superalgebra with dimension 2|2 to be a quadratic Lie superalgebra. [ABSTRACT FROM AUTHOR]

Using the approach of Gautam and Toledano Laredo, we construct an explicit isomorphism of the Yangian Yħ(A(m, n)) of the special linear Lie superalgebra and the quantum loop superalgebra Uħ(LA(m, n)). [ABSTRACT FROM AUTHOR]

We review recent results obtained at the intersection of the theory of quantum deformed Calogero-Moser-Sutherland systems and the theory of Lie superalgebras. We begin with a definition of admissible deformations of root systems of basic classical Lie superalgebras. For classical series, we prove the existence of Lax pairs. Connections between infinite-dimensional Calogero-Moser-Sutherland systems, deformed quantum CMS systems, and representation theory of Lie superalgebras are discussed. [ABSTRACT FROM AUTHOR]

By using the idea of Wakimoto’s free field, we construct a class of representations for the Lie superalgebra D(2, 1; α) on the tensor product of a polynomial algebra and an exterior algebra involving one parameter λ. Then we obtain the necessary and sufficient condition for the representations to be irreducible. In fact, the representation is irreducible if and only if the parameter λ satisfies (λ+m)(λ−1+ααm)≠0 for any m ∈ ℤ+. [ABSTRACT FROM AUTHOR]

We review the list of non-degenerate invariant (super)symmetric bilinear forms (briefly: NIS) on the following simple (relatives of) Lie (super)algebras: (a) with symmetrizable Cartan matrix of any growth, (b) with non-symmetrizable Cartan matrix of polynomial growth, (c) Lie (super)algebras of vector fields with polynomial coefficients, (d) stringy a.k.a. superconformal superalgebras, (e) queerifications of simple restricted Lie algebras. Over algebraically closed fields of positive characteristic, we establish when the deform (i.e., the result of deformation) of the known finite-dimensional simple Lie (super)algebra has a NIS. Amazingly, in most of the cases considered, if the Lie (super)algebra has a NIS, its deform has a NIS with the same Gram matrix after an identification of bases of the initial and deformed algebras. We do not consider odd parameters of deformations. Closely related with simple Lie (super)algebras with NIS is the notion of doubly extended Lie (super)algebras of which affine Kac-Moody (super)algebras are the most known examples. [ABSTRACT FROM AUTHOR]

We construct the graded triple Lie commutator of cubic supermatrices, which we call the quantum super Nambu bracket of cubic supermatrices, and prove that it satisfies the graded Filippov-Jacobi identity of 3-Lie superalgebra. For this purpose we use the basic notions of the calculus of 3-dimensional matrices, define the Z2-graded (or super) structure of a cubic matrix relative to one of the directions of a cubic matrix and the super trace of a cubic supermatrix. Making use of the super trace of a cubic supermatrix we introduce the triple product of cubic supermatrices and find the identities for this triple product, where one of them can be regarded as the analog of ternary associativity. We also show that given a Lie algebra one can construct the n-ary Lie bracket by means of an (n-2)-cochain of given Lie algebra and find the conditions under which this n-ary bracket satisfies the Filippov-Jacobi identity, thereby inducing the structure of n-Lie algebra. We extend this approach to n-Lie superalgebras. [ABSTRACT FROM AUTHOR]

In this article we construct link invariants and 3-manifold invariants from the quantum group associated with the Lie superalgebra sl(2|1). The construction is based on nilpotent irreducible finite dimensional representations of quantum group Uξsl(2|1) where ξ is a root of unity of odd order. These constructions use the notion of modified trace and relative G-modular category of previous authors. [ABSTRACT FROM AUTHOR]

We give explicit formulas proving that the following Lie (super)algebras are restricted: known exceptional simple vectorial Lie (super)algebras in characteristic 3, deformed Lie (super)algebras with indecomposable Cartan matrix, simple subquotients over an algebraically closed field of characteristic 3 of these (super)algebras (under certain conditions), and deformed divergence-free Lie superalgebras of a certain type with any finite number of indeterminates in any characteristic. [ABSTRACT FROM AUTHOR]

We develop a biorthogonal formalism for non-Hermitian multimode and multiphoton Jaynes-Cummings models. For these models, we define supersymmetric generators, which are especially convenient for diagonalizing the Hamiltonians. The Hamiltonian and its adjoint are expressed in terms of supersymmetric generators having the Lie superalgebra properties. The method consists in using a similarity dressing operator that maps onto spaces suitable for diagonalizing Hamiltonians even in an infinite-dimensional Hilbert space. We then successfully solve the eigenproblems related to the Hamiltonian and its adjoint. For each model, the eigenvalues are real, while the eigenstates do not form a set of orthogonal vectors. We then introduce the biorthogonality formalism to construct a consistent theory. [ABSTRACT FROM AUTHOR]

We give a proof of a Schur-Weyl duality statement between the Brauer algebra and the ortho-symplectic Lie superalgebra $$\mathfrak {osp}(V)$$ . [ABSTRACT FROM AUTHOR]

Let $$\mathcal {O}^\mathrm{int}_q(m|n)$$ be a semisimple tensor category of modules over a quantum ortho-symplectic superalgebra of type B, C, D introduced in Kwon (Int Math Res Not, 2015. doi:). It is a natural counterpart of the category of finitely dominated integrable modules over a quantum group of type B, C, D from a viewpoint of super duality. Continuing the previous work on type B and C (Kwon in Int Math Res Not, 2015. doi:), we classify the irreducible modules in $$\mathcal {O}^\mathrm{int}_q(m|n)$$ and prove the existence and uniqueness of their crystal bases in case of type D. A new combinatorial model of classical crystals of type D is introduced, whose super analog gives a realization of crystals for the highest weight modules in $$\mathcal {O}^\mathrm{int}_q(m|n)$$ . [ABSTRACT FROM AUTHOR]

We consider the action of the Lie superalgebra $${\mathcal {K}}(2)$$ by Lie derivative on the superspace of linear differential operators acting on the direct sum of the superspaces of weighted densities $${\mathfrak {S}}^{n}_{\beta }=\bigoplus _{k=0}^{2n}{\mathfrak {F}}^2_{\beta -\frac{k}{2}}$$ . We study the generic formal deformations of this action that become trivial once restricted to the orthosymplectic Lie superalgebra $${\mathfrak {osp}}(2|2)$$ . Necessary and sufficient conditions for integrability of any infinitesimal deformations of $${\mathfrak {S}}^{n}_{\beta }$$ are given. We describe completely the formal deformations for some spaces $${\mathfrak {S}}^{n}_{\beta }$$ and we give concrete examples of non trivial deformations. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the concept of normed Lie superalgebras and define the superhomomorphism and the superderivation in normed Lie superalgebras. We define a generalized T-orbitally complete metric space and prove a new fixed point alternative concerning the stability problem of functional equations. Consequently, we deal with the stability of the superhomomorphism and the superderivation in Lie superalgebras using that new fixed point result. In addition, we find some conditions under which an approximate superhomomorphism or superderivation is an exact superhomomorphism or superderivation. [ABSTRACT FROM AUTHOR]

We study ℤ-graded identities of Lie superalgebras of the type b( t), t ≥ 2, over a field of characteristic zero. Our main result is that the n-th codimension is strictly less than $(\dim b(t))^n$ asymptotically. As a consequence we obtain an upper bound for ordinary (non-graded) PI-exponent for each simple Lie superalgebra b( t), t ≥ 3. [ABSTRACT FROM AUTHOR]

Over the $$(1, n)$$ -dimensional real superspace, we consider the Lie superalgebra $$\mathcal {K}(n)$$ of contact vector fields on $${\mathbb {R}}^{1|n}$$ . We classify $$\mathfrak {osp}(n|2)$$ -invariant linear differential operators acting on the superspaces of weighted densities. This result allows us to compute the first differential cohomology of $$\mathcal {K}(n)$$ with coefficients in the superspace of weighted densities, vanishing on the Lie superalgebra $$\mathfrak {osp}(n|2)$$ . We explicitly give 1-cocycles spanning these cohomology spaces. [ABSTRACT FROM AUTHOR]

Suppose a group $$\Gamma $$ acts on a scheme $$X$$ and a Lie superalgebra $$\mathfrak {g}$$ . The corresponding equivariant map superalgebra is the Lie superalgebra of equivariant regular maps from $$X$$ to $$\mathfrak {g}$$ . We classify the irreducible finite dimensional modules for these superalgebras under the assumptions that the coordinate ring of $$X$$ is finitely generated, $$\Gamma $$ is finite abelian and acts freely on the rational points of $$X$$ , and $$\mathfrak {g}$$ is a basic classical Lie superalgebra (or $$\mathfrak {sl}\,(n,n)$$ , $$n \ge 1$$ , if $$\Gamma $$ is trivial). We show that they are all (tensor products of) generalized evaluation modules and are parameterized by a certain set of equivariant finitely supported maps defined on $$X$$ . Furthermore, in the case that the even part of $$\mathfrak {g}$$ is semisimple, we show that all such modules are in fact (tensor products of) evaluation modules. On the other hand, if the even part of $$\mathfrak {g}$$ is not semisimple (more generally, if $$\mathfrak {g}$$ is of type I), we introduce a natural generalization of Kac modules and show that all irreducible finite dimensional modules are quotients of these. As a special case, our results give the first classification of the irreducible finite dimensional modules for twisted loop superalgebras. [ABSTRACT FROM AUTHOR]

Superderivations for the eight families of finite or infinite dimensional graded Lie superalgebras of Cartan-type over a field of characteristic p > 3 are completely determined by a uniform approach: The infinite dimensional case is reduced to the finite dimensional case and the latter is further reduced to the restrictedness case, which proves to be far more manageable. In particular, the structures and dimension formulas are clearly described for the outer superderivation algebras of those Lie superalgebras. Certain known results are also covered. [ABSTRACT FROM AUTHOR]

The principal filtration of the infinite-dimensional odd contact Lie superalgebra over a field of characteristic p > 2 is proved to be invariant under the automorphism group by investigating ad-nilpotent elements and determining certain invariants such as subalgebras generated by some ad-nilpotent elements. Then, it is proved that two automorphisms coincide if and only if they coincide on the −1 component with respect to the principal grading. Finally, all the odd contact superalgebras are classified up to isomorphisms. [ABSTRACT FROM AUTHOR]

We consider symmetric pairs of Lie superalgebras and introduce a graded Harish-Chandra homomorphism. Generalising results of Harish-Chandra and V. Kac, M. Gorelik, we prove that, assuming reductivity, its image is a certain explicit filtered subalgebra J( $ \mathfrak{a} $) of the Weyl invariants on a Cartan subspace whose associated graded gr J( $ \mathfrak{a} $) is the image of Chevalley's restriction map on symmetric invariants. In contrast to the known cases, J( $ \mathfrak{a} $) is in general not isomorphic to gr J( $ \mathfrak{a} $). [ABSTRACT FROM AUTHOR]

We provide formulas for the denominator and superdenominator of a basic classical type Lie superalgebra for any set of positive roots. We establish a connection between certain sets of positive roots and the theory of reductive dual pairs of real Lie groups, and, as an application of these formulas, we recover the Theta correspondence for compact dual pairs. Along the way we give an explicit description of the real forms of basic classical type Lie superalgebras. [ABSTRACT FROM AUTHOR]

Let g be a finite dimensional special odd Lie superalgebra over an algebraically closed field $\mathbb{F}$ of characteristic p > 3. The sufficient and necessary condition is given for g possessing a nondegenerate associative form and in this case the second cohomology group H( g, $H^2 (\mathfrak{g},\mathbb{F})$) is completely determined. [ABSTRACT FROM AUTHOR]

Lie antialgebras is a class of supercommutative algebras recently appeared in symplectic geometry. We define the notion of enveloping algebra of a Lie antialgebra and study its properties. We show that every Lie antialgebra is canonically related to a Lie superalgebra and prove that its enveloping algebra is a quotient of the enveloping algebra of the corresponding Lie superalgebra. [ABSTRACT FROM AUTHOR]

We classify deformations of the standard embedding of the Lie superalgebra $$ \mathcal{K} $$ (2) of contact vector fields on the (1, 2)-dimensional supercircle into the Lie superalgebra SΨD(S 1|2) of pseudodifferential operators on the supercircle S 1|2. The proposed approach leads to the deformations of the central charge induced on $$ \mathcal{K} $$ (2) by the canonical central extension of SΨD(S 1|2). [ABSTRACT FROM AUTHOR]

We classify deformations of the standard embedding of the Lie superalgebra $$ \mathcal{K} $$ (2) of contact vector fields on the (1, 2)-dimensional supercircle into the Lie superalgebra SΨD(S 1|2) of pseudodifferential operators on the supercircle S 1|2. The proposed approach leads to the deformations of the central charge induced on $$ \mathcal{K} $$ (2) by the canonical central extension of SΨD(S 1|2). [ABSTRACT FROM AUTHOR]

A super-soliton hierarchy and its super-Hamiltonian structure is obtained respectively based on Lie super-algebra and associated super-trace identity. [ABSTRACT FROM AUTHOR]

We consider the δ-derivations of classical Lie superalgebras and prove that these superalgebras admit nonzero δ-derivations only when δ = 0, ½, 1. The structure of ½-derivations for classical Lie superalgebras is completely determined. [ABSTRACT FROM AUTHOR]

It is proved that the exponential growth rate of identities in any superalgebra with a nilpotent commutator subalgebra over a field of zero characteristic is an integer. [ABSTRACT FROM AUTHOR]

For the Kac-Moody superalgebra associated with the loop superalgebra with values in a finite-dimensional Lie superalgebra g, we show what its quadratic Casimir element is equal to if the Casimir element for g is known (if g has an even invariant supersymmetric bilinear form). The main tool is the Wick normal form of the even quadratic Casimir operator for the Kac-Moody superalgebra associated with g; this Wick normal form is independently interesting. If g has an odd invariant supersymmetric bilinear form, then we compute the cubic Casimir element. In addition to the simple Lie superalgebras g = g(A) with a Cartan matrix A for which the Shapovalov determinant was known, we consider the Poisson Lie superalgebra poi(0|n) and the related Kac-Moody superalgebra. [ABSTRACT FROM AUTHOR]

The purpose of this contribution, is to initiate a classification of Lie superalgebras (LS) of dimension five, over the base field ℝ of real numbers. We use the “ graded skew-symmetry” and the “ graded Jacobi identity” in order to get restrictions for the commutators and anticommutators of an arbitrary five-dimensional Lie superalgebra L = L 0⊕ L 1 [ABSTRACT FROM AUTHOR]

We construct a basis for a free Lie superalgebra consisting of right-normed words $$[a_{i_1 } [a_{i_2 } [...[a_{i_{t - 1} } a_{i_t } ]...]]]$$ , where $$a_{i_j } $$ are free generators. [ABSTRACT FROM AUTHOR]

In this Letter, we will prove a super-analogue of a well-known result by Kontsevich which states that the homology of a certain complex generated by isomorphism classes of oriented graphs can be calculated as the Lie algebra homology of an infinite-dimensional Lie algebra of symplectic vector fields. [ABSTRACT FROM AUTHOR]

We give a representation-theoretical interpretation of the Gelfand–Tsetlin pattern for strict partitions. Using the Howe duality involving a pair of the queer Lie superalgebras and an analogue of the Littlewood–Richardson rule for Schur Q-functions, we show that such patterns give the branching rule for the irreducible tensor representations of the queer Lie superalgebra. [ABSTRACT FROM AUTHOR]