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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]

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]

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]

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]

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 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]

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]

Inspired by [12] , we give shuffle algebra realization of positive part of quantum affine superalgebra U v (D ˆ (2 , 1 ; θ)) associated to any simple root systems. We also determine the shuffle algebra associated to sl ˆ (2 | 1) with odd root system when v is a primitive root of unity of even order, generalizing results in [4]. [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]

The inverses of indecomposable Cartan matrices are computed for finite-dimensional Lie algebras and Lie superalgebras over fields of any characteristic, and for hyperbolic (almost affine) complex Lie (super)algebras. This enables one to express the fundamental weights in terms of simple roots corresponding to the Cartan matrix. We discovered three yet inexplicable new phenomena, of which (a) and (b) concern hyperbolic (almost affine) complex Lie (super)algebras: (a) several of the inverses of Cartan matrices have all their elements negative (not just non-positive, as they should be according to an a priori characterization due to Zhang Hechun); (b) the 0s only occur on the main diagonals of the inverses; (c) the determinants of inequivalent Cartan matrices of the simple Lie (super)algebra of any dimension may differ (in any characteristic). We interpret most of the results of Wei and Zou (2017) [31] as inverses of the Gram matrices of non-degenerate invariant symmetric bilinear forms on the (super)algebras considered, not of Cartan matrices, and give more adequate references. In particular, the inverses of Cartan matrices of simple Lie algebras were already published, starting with Dynkin's paper in 1952, see also Table 2 in Springer's book by Onishchik and Vinberg (1990). [ABSTRACT FROM AUTHOR]

Reduction algebras are known by many names in the literature, including step algebras, Mickelsson algebras, Zhelobenko algebras, and transvector algebras, to name a few. These algebras, realized by raising and lowering operators, allow for the calculation of Clebsch-Gordan coefficients, branching rules, and intertwining operators; and have connections to extremal equations and dynamical R-matrices in integrable face models. In this paper we continue the study of the diagonal reduction superalgebra A of the orthosymplectic Lie superalgebra osp (1 | 2). We construct a Harish-Chandra homomorphism, Verma modules, and study the Shapovalov form on each Verma module. Using these results, we prove that the ghost center (center plus anti-center) of A is generated by two central elements and one anti-central element (analogous to the Scasimir due to Leśniewski for osp (1 | 2)). As another application, we classify all finite-dimensional irreducible representations of A. Lastly, we calculate an infinite-dimensional tensor product decomposition explicitly. [ABSTRACT FROM AUTHOR]

The classification of simple structurable superalgebras of Cartan type over an algebraically closed field of characteristic 0 has been done recently by Pozhidaev and Shestakov [8]. Here we present an easy construction and quick classification of the algebras using a superalgebra version of results due to Elduque and Okubo [5] relating Lie algebras with S4-actions and structurable algebras. [ABSTRACT FROM AUTHOR]

A super-Burgers hierarchy and its super-Hamiltonian structure is obtained respectively based on Lie super-algebra and is associated with super-trace identity. [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 study various specializations of the colored HOMFLY-PT polynomial. These specializations are used to show that the multivariable link invariants arising from a complex family of 픰픩(m|n) super-modules previously defined by the authors contain both the multivariable Alexander polynomial and Kashaev's invariants. We conjecture these multivariable link invariants also specialize to the generalized multivariable Alexander invariants defined by Y. Akutsu, T. Deguchi, and T. Ohtsuki. [ABSTRACT FROM AUTHOR]

Abstract: In this paper we use the Etingof–Kazhdan quantization of Lie bi-superalgebras to investigate some interesting questions related to Drinfeld–Jimbo type superalgebra associated to a Lie superalgebra of classical type. It has been shown that the D-J type superalgebra associated to a Lie superalgebra of type A-G, with the distinguished Cartan matrix, is isomorphic to the E-K quantization of the Lie superalgebra. The first main result in the present paper is to extend this to arbitrary Cartan matrices. This paper also contains two other main results: (1) a theorem stating that all highest weight modules of a Lie superalgebra of type A-G can be deformed to modules over the corresponding D-J type superalgebra and (2) a super version of the Drinfeld–Kohno theorem. [Copyright &y& Elsevier]

Abstract: For every semi-simple Lie algebra one can construct the Drinfeld–Jimbo algebra . This algebra is a deformation Hopf algebra defined by generators and relations. To study the representation theory of , Drinfeld used the KZ-equations to construct a quasi-Hopf algebra . He proved that particular categories of modules over the algebras and are tensor equivalent. Analogous constructions of the algebras and exist in the case when is a Lie superalgebra of type A-G. However, Drinfeld''s proof of the above equivalence of categories does not generalize to Lie superalgebras. In this paper, we will discuss an alternate proof for Lie superalgebras of type A-G. Our proof utilizes the Etingof–Kazhdan quantization of Lie (super)bialgebras. It should be mentioned that the above equivalence is very useful. For example, it has been used in knot theory to relate quantum group invariants and the Kontsevich integral. [Copyright &y& Elsevier]

The finite-dimensional odd contact Lie superalgebras KO ( n , n + 1, t ) over a field of prime characteristic are studied, where n is a positive integer and t is an n -tuple of non-negative integers. In particular, it is proven that KO ( n , n + 1, t ) is simple and has no non-singular associative bilinear forms. Moreover, an explicit description of the derivation superalgebra of KO ( n , n + 1, t ) is given, and as a consequence it is shown that the outer derivation superalgebra of KO ( n , n + 1, t ) is Abelian of dimension . Communicated by K. Misra. [ABSTRACT FROM AUTHOR]

Abstract: A noncommutative Jordan algebra of a specific type is attached to any -balanced Freudenthal Kantor triple system, in such a way that the triple product in this system is determined by the binary product in the algebra. Over fields of characteristic zero, the simple noncommutative Jordan algebras of this type are classified. [Copyright &y& Elsevier]

In this paper, we obtain the Lie superalgebra of superderivations of H type Lie superalgebra and discuss the relation between skew superderivations space SkDer(L) and central extension H2(L, F) on some Lie superalgebra L. As an application, we obtain the dimension of central extension of Lie superalgebras with a nondegenerate associative forms. [ABSTRACT FROM AUTHOR]

In the present paper, we give the derivation algebra Der g and the holomorph h(g) of the finite dimensional Heisenberg superalgebra g over the complex field C. We also give the derivation algebra Der h(g) of h(g). We prove that Der g is a simple complete Lie superalgebra, h(g) is not a complete Lie superalgebra, but its derivation algebra Der h(g) is a simple complete Lie superalgebra. [Copyright &y& Elsevier]