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We construct universal monoidal categories of topological tensor supermodules over the Lie superalgebras gl (V ⊕ Π V) and osp (V ⊕ Π V) associated with a Tate space V. Here V ⊕ Π V is a Z / 2 Z -graded topological vector space whose even and odd parts are isomorphic to V. We discuss the purely even case first, by introducing monoidal categories T ˆ gl (V) , T ˆ o (V) and T ˆ sp (V) , and show that these categories are anti-equivalent to respective previously studied categories T gl (V) , T o (V) , T sp (V). These latter categories have certain universality properties as monoidal categories, which consequently carry over to T ˆ gl (V) , T ˆ o (V) and T ˆ sp (V). Moreover, the categories T o (V) and T sp (V) are known to be equivalent, and this implies the equivalence of the categories T ˆ o (V) and T ˆ sp (V). After introducing a supersymmetric setting, we establish the equivalence of the category T ˆ gl (V) with the category T ˆ gl (V ⊕ Π V) , and the equivalence of both categories T ˆ o (V) and T ˆ sp (V) with T ˆ osp (V ⊕ Π V). [ABSTRACT FROM AUTHOR]

In this paper, we propose the notion of a transposed Poisson superalgebra. We prove that a transposed Poisson superalgebra can be constructed by means of a commutative associative superalgebra and an even degree derivation of this algebra. Making use of this, we construct two examples of the transposed Poisson superalgebra. One of them is the graded differential algebra of differential forms on a smooth finite dimensional manifold, where we use the Lie derivative as an even degree derivation. The second example is the commutative superalgebra of basic fields of the quantum YangâMills theory, where we use the BRST-supersymmetry as an even degree derivation to define a graded Lie bracket. We show that a transposed Poisson superalgebra has six identities that play an important role in the study of the structure of this algebra.

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]

In this paper, we propose the notion of a transposed Poisson superalgebra.We prove that a transposed Poisson superalgebra can be constructed by means of a commutative associative superalgebra and an even degree derivation of this algebra. Making use of this, we construct two examples of the transposed Poisson superalgebra. One of them is the graded differential algebra of differential forms on a smooth finite dimensional manifold, where we use the Lie derivative as an even degree derivation. The second example is the commutative superalgebra of basic fields of the quantum Yang--Mills theory, where we use the BRST-supersymmetry as an even degree derivation to define a graded Lie bracket. We show that a transposed Poisson superalgebra has six identities that play an important role in the study of the structure of this algebra. [ABSTRACT FROM AUTHOR]

This paper performed an investigation into the s-embedding of the Lie superalgebra (→ S 1 ∣ 1 ) , a representation of smooth vector fields on a (1,1)-dimensional super-circle. Our primary objective was to establish a precise definition of the s-embedding, effectively dissecting the Lie superalgebra into the superalgebra of super-pseudodifferential operators ( S ψ D ⊙ ) residing on the super-circle S 1 | 1 . We also introduce and rigorously define the central charge within the framework of (→ S 1 ∣ 1 ) , leveraging the canonical central extension of S ψ D ⊙ . Moreover, we expanded the scope of our inquiry to encompass the domain of fuzzy Lie algebras, seeking to elucidate potential connections and parallels between these ostensibly distinct mathematical constructs. Our exploration spanned various facets, including non-commutative structures, representation theory, central extensions, and central charges, as we aimed to bridge the gap between Lie superalgebras and fuzzy Lie algebras. To summarize, this paper is a pioneering work with two pivotal contributions. Initially, a meticulous definition of the s-embedding of the Lie superalgebra (→ S 1 | 1 ) is provided, emphasizing the representationof smooth vector fields on the (1,1)-dimensional super-circle, thereby enriching a fundamental comprehension of the topic. Moreover, an investigation of the realm of fuzzy Lie algebras was undertaken, probing associations with conventional Lie superalgebras. Capitalizing on these discoveries, we expound upon the nexus between central extensions and provide a novel deformed representation of the central charge. [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]

Abstract The present paper is devoted to studying 2-local superderivations on the twisted N = 2 superconformal algebra, the Ramond N = 2 super-Virasoro algebra and the super W -algebra S W ( 3 2 , 3 2 ) . It is proved that all 2-local superderivations on these Lie superalgebras are superderivations. [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 this note, the compatible left-symmetric superalgebra structures on an infinite-dimensional Lie superalgebra with some natural grading conditions are completely determined. The results of earlier work on left-symmetric algebra structures on the Virasoro algebra play an essential role in determining these compatible structures. [ABSTRACT FROM AUTHOR]

Copyright of Journal of Jilin University (Science Edition) / Jilin Daxue Xuebao (Lixue Ban) is the property of Zhongguo Xue shu qi Kan (Guang Pan Ban) Dian zi Za zhi She and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

We propose a method for generating higher-dimensional nonisospectral super integrable coupling hierarchies associated with a new type of higher-dimensional Lie superalgebra. As applications, we introduce a coupled and a multi-component super Dirac spectral problems respectively, and thus deduce a coupled generalized super integrable Dirac hierarchy and a multi-component generalized super integrable Dirac hierarchy. Additionally, we discuss the super Hamiltonian structures of the resulting higher-dimensional super Dirac hierarchies.

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]

This paper is aimed at determining the derivation superalgebra of modular Lie superalgebra $ \overline{K}(n, m) $. To that end, we first describe the $ \mathbb{Z} $-homogeneous derivations of $ \overline{K}(n, m) $. Then we obtain the derivation superalgebra $ Der(\overline{K}) $. Finally, we partly determine the derivation superalgebra $ Der(K) $ by virtue of the invariance of $ K(n, m) $ under $ Der(\overline{K}) $.

Let be the planar Galilean conformal superalgebra. In this paper, we determine all the super-skewsymmetric super-biderivations of . We find that every super-skewsymmetric super-biderivation of the planar Galilean conformal superalgebra is inner. [ABSTRACT FROM AUTHOR]

We explicitly describe the second cohomology of the Lie superalgebra 풦 ⁢ (1) of contact vector fields on the supercircle S 1 | 1 with coefficients in the spaces of weighted densities. We deduce the second cohomology of 풦 ⁢ (1) with coefficients in the Poisson algebra of pseudodifferential symbols on S 1 | 1 . We study formal deformations of the standard embedding of 풦 ⁢ (1) into the Poisson superalgebra of pseudodifferential symbols. [ABSTRACT FROM AUTHOR]

We explicitly describe the second cohomology of the Lie superalgebra 풦 ⁢ (1) of contact vector fields on the supercircle S 1 | 1 with coefficients in the spaces of weighted densities. We deduce the second cohomology of 풦 ⁢ (1) with coefficients in the Poisson algebra of pseudodifferential symbols on S 1 | 1 . We study formal deformations of the standard embedding of 풦 ⁢ (1) into the Poisson superalgebra of pseudodifferential symbols. [ABSTRACT FROM AUTHOR]

This paper performed an investigation into the s-embedding of the Lie superalgebra (→S1∣1), a representation of smooth vector fields on a (1,1)-dimensional super-circle. Our primary objective was to establish a precise definition of the s-embedding, effectively dissecting the Lie superalgebra into the superalgebra of super-pseudodifferential operators ( SψD⊙) residing on the super-circle S1|1. We also introduce and rigorously define the central charge within the framework of (→S1∣1), leveraging the canonical central extension of SψD⊙. Moreover, we expanded the scope of our inquiry to encompass the domain of fuzzy Lie algebras, seeking to elucidate potential connections and parallels between these ostensibly distinct mathematical constructs. Our exploration spanned various facets, including non-commutative structures, representation theory, central extensions, and central charges, as we aimed to bridge the gap between Lie superalgebras and fuzzy Lie algebras. To summarize, this paper is a pioneering work with two pivotal contributions. Initially, a meticulous definition of the s-embedding of the Lie superalgebra (→S1|1) is provided, emphasizing the representationof smooth vector fields on the (1,1)-dimensional super-circle, thereby enriching a fundamental comprehension of the topic. Moreover, an investigation of the realm of fuzzy Lie algebras was undertaken, probing associations with conventional Lie superalgebras. Capitalizing on these discoveries, we expound upon the nexus between central extensions and provide a novel deformed representation of the central charge.

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]

We proceed to generalize the Yang-Baxter (YB) deformation of Wess-Zumino-Witten (WZW) model to the Lie supergroups case. This generalization enables us to utilize various kinds of solutions of the (modified) graded classical Yang-Baxter equation ((m)GCYBE) to classify the YB deformations of WZW models based on the Lie supergroups. We obtain the inequivalent solutions (classical r-matrices) of the (m)GCYBE for the gl(1|1) and (C3+A) Lie superalgebras in the non-standard basis, in such a way that the corresponding automorphism transformations are employed. Then, the YB deformations of the WZW models based on the GL(1|1) and (C3+A) Lie supergroups are specified by skew-supersymmetric classical r-matrices satisfying (m)GCYBE. In some cases for both families of deformed models, the metrics remain invariant under the deformation, while the components of B-fields are changed. After checking the conformal invariance of the models up to one-loop order, it is concluded that the GL(1|1) and (C3+A) WZW models are conformal theories within the classes of the YB deformations preserving the conformal invariance. However, our results are interesting in themselves, but at a constructive level, may prompt many new insights into (generalized) supergravity solutions.

To any Frobenius superalgebra A we associate an oriented Frobenius Brauer category and an affine oriented Frobenius Brauer category. We define natural actions of these categories on categories of supermodules for general linear Lie superalgebras g l m | n (A) with entries in A. These actions generalize those on module categories for general linear Lie superalgebras and queer Lie superalgebras, which correspond to the cases where A is the ground field and the two-dimensional Clifford superalgebra, respectively. [ABSTRACT FROM AUTHOR]

For classical Lie superalgebras of type I, we provide necessary and sufficient conditions for a Verma supermodule $\Delta (\lambda)$ to be such that every nonzero homomorphism from another Verma supermodule to $\Delta (\lambda)$ is injective. This is applied to describe the socle of the cokernel of an inclusion of Verma supermodules over the periplectic Lie superalgebras $\mathfrak {pe} (n)$ and, furthermore, to reduce the problem of description of $\mathrm {Ext}^1_{\mathcal O}(L(\mu),\Delta (\lambda))$ for $\mathfrak {pe} (n)$ to the similar problem for the Lie algebra $\mathfrak {gl}(n)$. Additionally, we study the projective and injective dimensions of structural supermodules in parabolic category $\mathcal O^{\mathfrak {p}}$ for classical Lie superalgebras. In particular, we completely determine these dimensions for structural supermodules over the periplectic Lie superalgebra $\mathfrak {pe} (n)$ and the orthosymplectic Lie superalgebra $\mathfrak {osp}(2|2n)$. [ABSTRACT FROM AUTHOR]

In this paper we give an in-deph analysis of the nilpotency index of nilpotent homogeneous inner superderivations in associative prime superalgebras with and without superinvolution. We also present examples of all the different cases that our analysis exhibits for the nilpotency indices of the inner superderivations. [ABSTRACT FROM AUTHOR]

Bol operators (Bols for short) are differential operators invariant under the projective action of (2) ≃ (2) between spaces of weighted densities on the 1-dimensional manifold. Here, we described analogs of Bols: (a + 1 | b) -invariant differential operators between spaces of tensor fields on (a | b) -dimensional supermanifolds with irreducible, as (a | b) -modules, fibers of arbitrary, even infinite, dimension for certain "key" values of a and b — the ones for which the solution is describable. We discovered many new operators for (a | b) = (2 | 0) , (0 | 3) and for the case of 1 | 1 -dimensional general superstring, which looks like a most natural superization of Bol's result, additional to the cases of super analogs of Bols between spaces of weighted densities on the 1 | n -dimensional superstrings with a contact structure we classified in arXiv:2110.10504. In the case of fibers of dimension > 1 , there are (a + b − 1) -parameter families of Bols, whereas there are no non-scalar nonzero differential operators between spaces of weighted densities. These two extreme answers justify the selection of cases here. [ABSTRACT FROM AUTHOR]

This paper is to study vertex operator superalgebras which are strongly generated by their weight-2 and weight- 3 2 homogeneous subspaces. Among the main results, it is proved that if such a vertex operator superalgebra V is simple, then V (2) has a canonical commutative associative algebra structure equipped with a non-degenerate symmetric associative bilinear form and V (3 2) is naturally a V (2) -module equipped with a V (2) -valued symmetric bilinear form and a non-degenerate (C -valued) symmetric bilinear form, satisfying a set of conditions. On the other hand, assume that A is any commutative associative algebra equipped with a non-degenerate symmetric associative bilinear form and assume that U is an A -module equipped with a symmetric A -valued bilinear form and a non-degenerate (C -valued) symmetric bilinear form, satisfying the corresponding conditions. Then we construct a Lie superalgebra L (A , U) and a simple vertex operator superalgebra L L (A , U) (ℓ , 0) for every nonzero number ℓ such that L L (A , U) (ℓ , 0) (2) = A and L L (A , U) (ℓ , 0) (3 2) = U. [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 is well-known, the dimension of the space spanned by the non-degenerate invariant symmetric bilinear forms (NISes) on any simple finite-dimensional Lie algebra or Lie superalgebra is equal to at most 1 if the characteristic of the algebraically closed ground field is not 2. We prove that in characteristic 2, the superdimension of the space spanned by NISes can be equal to 0, or 1, or 0 | 1 , or 1 | 1 ; it is equal to 1 | 1 if and only if the Lie superalgebra is a queerification (defined in arXiv:1407.1695) of a simple classically restricted Lie algebra with a NIS (for examples, mainly in characteristic ≠2, see arXiv:1806.05505). We give examples of NISes on deformations (with both even and odd parameters) of several simple finite-dimensional Lie superalgebras in characteristic 2. We also recall examples of multiple NISes on simple Lie algebras over non-closed fields. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the notions of non-abelian tensor and exterior products of two ideal graded crossed submodules of a given crossed module of Lie superalgebras. We also study some of their basic properties and their connection with the second homology of crossed modules of Lie superalgebras. [ABSTRACT FROM AUTHOR]

The present article is devoted to studying 2-Local superderivations on the deformative super W-algebras W λ s (2,2). It is proved that all 2-Local superderivations on the deformative super W-algebras W λ s (2,2) are superderivations. [ABSTRACT FROM AUTHOR]

We study a class of modules, called Chari-Venkatesh modules, for the current superalgebra sl (1 | 2) [ t ]. This class contains other important modules, such as graded local Weyl, truncated local Weyl and Demazure-type modules. We prove that Chari-Venkatesh modules can be realized as fusion products of generalized Kac modules. In particular, this proves Feigin and Loktev's conjecture, that fusion products are independent of their fusion parameters, in the case where the fusion factors are generalized Kac modules. As an application of our results, we obtain bases, dimension and character formulas for Chari-Venkatesh modules. [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 define the current Lie superalgebra L ⊗ A , where L is a Lie superalgebra and A is an associative commutative algebra with unity. we proved every skew-symmetric super-biderivation on L ⊗ A is of the form of the centroid, if the same holds true for L. We also proved every linear super-commuting map on L ⊗ A belong to Cent( L ⊗ A ), if the same holds true for L. [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]

The Bol operators are unary differential operators between spaces of weighted densities on the 1-dimensional manifold invariant under projective transformations of the manifold. On the 1 | n -dimensional supermanifold (superstring) ℳ , we classify analogs of Bol operators invariant under the simple maximal subalgebra of the same rank as its simple ambient superalgebra of vector fields on ℳ and containing all elements of negative degree of in a ℤ -grading. We also consider the Lie superalgebras of vector fields preserving a contact structure on the superstring ℳ. We have discovered many new operators. [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]

The Grigorchuk and Gupta–Sidki groups play a fundamental role in modern group theory. They are natural examples of self-similar finitely generated periodic groups. The author constructed their analogue in case of restricted Lie algebras of characteristic 2 [V. M. Petrogradsky, Examples of self-iterating Lie algebras, J. Algebra302(2) (2006) 881–886], Shestakov and Zelmanov extended this construction to an arbitrary positive characteristic [I. P. Shestakov and E. Zelmanov, Some examples of nil Lie algebras, J. Eur. Math. Soc. (JEMS)10(2) (2008) 391–398]. Now, we construct a family of so called clover 3-generated restricted Lie algebras T (Ξ) , where a field of positive characteristic is arbitrary and Ξ an infinite tuple of positive integers. All these algebras have a nil p -mapping. We prove that 1 ≤ GKdim T (Ξ) ≤ 3. We compute Gelfand–Kirillov dimensions of clover restricted Lie algebras with periodic tuples and show that these dimensions for constant tuples are dense on [ 1 , 3 ]. We construct a subfamily of nil restricted Lie algebras T (Ξ q , κ) , with parameters q ∈ ℕ , κ ∈ ℝ + , having extremely slow quasi-linear growth of type: γ T (Ξ q , κ) (m) = m ln ⋯ ln ︸ q m κ + o (1) , as m → ∞. The present research is motivated by construction by Kassabov and Pak of groups of oscillating growth [M. Kassabov and I. Pak, Groups of oscillating intermediate growth. Ann. Math. (2)177(3) (2013) 1113–1145]. As an analogue, we construct nil restricted Lie algebras of intermediate oscillating growth in [V. Petrogradsky, Nil restricted Lie algebras of oscillating intermediate growth, preprint (2020), arXiv:2004.05157]. We call them Phoenix algebras because, for infinitely many periods of time, the algebra is "almost dying" by having a "quasi-linear" growth as above, for infinitely many n it has a rather fast intermediate growth of type exp (n / (ln n) λ) , for such periods the algebra is "resuscitating". The present construction of three-generated nil restricted Lie algebras of quasi-linear growth is an important part of that result, responsible for the lower quasi-linear bound in that construction. [ABSTRACT FROM AUTHOR]

Let Ḡ be the simple algebraic supergroup SL (m | n) or (m | 2 n) over ℂ. Let = Lie (Ḡ) = 0 ̄ ⊕ 1 ̄ and let G = Ḡ (ℂ) , where ℂ is considered as a superalgebra concentrated in even degree. Suppose e ∈ 0 ̄ is nilpotent. We describe the centralizer e of e in and its center ( e). In particular, we give bases for e , ( e) and (( e)) G e . We also determine the labeled Dynkin diagram Δ with respect to e and subsequently describe the relation between (( e)) G e and Δ. [ABSTRACT FROM AUTHOR]

Copyright of Journal of Jilin University (Science Edition) / Jilin Daxue Xuebao (Lixue Ban) is the property of Zhongguo Xue shu qi Kan (Guang Pan Ban) Dian zi Za zhi She and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

The research is motivated by a construction of groups of oscillating growth by Kassabov and Pak [25] and a description of possible growth functions of finitely generated associative algebras by Bell and Zelmanov [9]. In this paper we address both, the question of possible growth functions in case of Lie algebras, and the Kurosh problem, because our examples of restricted Lie algebras have a nil p -mapping, which is an analogue of nillity for associative algebras or periodicity for groups. Namely, for any field of positive characteristic, we construct a family of 3-generated restricted Lie algebras of intermediate oscillating growth. We call them Phoenix algebras because, for infinitely many periods of time, the algebra is "almost dying" by having a quasi-linear growth, namely the lower Gelfand-Kirillov dimension is one, more precisely, the growth is of type n ( ln ⁡ ⋯ ln ︸ q times n) κ , where q ∈ N , κ > 0 are constants. On the other hand, for infinitely many n the growth function has a rather fast intermediate behavior of type exp ⁡ (n / (ln ⁡ n) λ) , λ being a constant determined by characteristic, for such periods the algebra is "resuscitating". Moreover, the growth function is bounded and oscillating between these two types of behavior. These restricted Lie algebras have a nil p -mapping, thus addressing the Kurosh problem as well. [ABSTRACT FROM AUTHOR]

The present paper is devoted to studying 2-local superderivations on the super Virasoro algebra and the super W (2 , 2) algebra. It is proved that all 2-local superderivations on the super Virasoro algebra as well as the super W (2 , 2) algebra are (global) superderivations. [ABSTRACT FROM AUTHOR]

In this article, we find a family S L n , m , in any arbitrary dimensions, of cohomologically rigid solvable Lie superalgebras with nilradical the model filiform Lie superalgebra L n , m . Moreover, we exhibit a family of cohomologically rigid solvable Lie superalgebras with nilradical the model nilpotent Lie superalgebra of generic characteristic sequence. Both cases correspond to solvable Lie superalgebras of maximal dimension for a given nilradical. Contrariwise, we will show that the family of Lie superalgebras S L n , m can be deformed if defined over a field of odd characteristic. [ABSTRACT FROM AUTHOR]

By constructing a new type of multi-component Lie superalgebra sl(2N,1), a method of generation of multi-component super integrable hierarchies is proposed. We discuss two applications and then obtain a coupled generalized super integrable Ablowitz–Kaup–Newell–Segur (AKNS) hierarchy and a multi-component generalized super integrable AKNS hierarchy respectively. Then, the super Hamiltonian structures of the resulting super integrable hierarchies are deduced by means of supertrace identity. • We construct a new type of multi-component Lie superalgebra. • A method of generation of multi-component super integrable hierarchies is proposed. • We obtain a coupled and a multi-component super integrable AKNS Hamiltonian hierarchies. [ABSTRACT FROM AUTHOR]

We propose a method for generating higher-dimensional nonisospectral super integrable coupling hierarchies associated with a new type of higher-dimensional Lie superalgebra. As applications, we introduce a coupled and a multi-component super Dirac spectral problems respectively, and thus deduce a coupled generalized super integrable Dirac hierarchy and a multi-component generalized super integrable Dirac hierarchy. Additionally, we discuss the super Hamiltonian structures of the resulting higher-dimensional super Dirac hierarchies. [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]

The generalized quantum group of type A is an affine analogue of the quantum group associated to a general linear Lie superalgebra, which appears in the study of solutions to the tetrahedron equation or the three-dimensional Yang--Baxter equation. In this paper we develop the crystal base theory for finite-dimensional representations of generalized quantum groups of type A. As the main result, we construct Kirillov--Reshetikhin modules, that is, a family of irreducible modules which have crystal bases. We also give an explicit combinatorial description of the crystal structure of Kirillov--Reshetikhin modules, the combinatorial R matrix and the energy function on their tensor products. [ABSTRACT FROM AUTHOR]

The main topic of this paper is to compute the first relative cohomology group of the Lie algebra of smooth vector fields Vect (R) , with coefficients in the space of trilinear differential operators that acts on tensor densities, D λ ¯ , μ , vanishing on the Lie algebra s l (2) , where λ ¯ = (λ 1 , λ 2 , λ 3) ∈ R 3 and μ ∈ R. [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]