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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]

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]

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]

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 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]

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]

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 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]

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]

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]

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]

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]