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We are concerned with finite-dimensional irreducible representations of the Yangians associated with the orthosymplectic Lie superalgebras osp 2 n + 1 | 2 m. Every such representation is highest weight and we use embedding theorems and odd reflections of Yangian type to derive necessary conditions for an irreducible highest weight representation to be finite-dimensional. We conjecture that these conditions are also sufficient. We prove the conjecture in the case n = 1 and arbitrary m ⩾ 1. [ABSTRACT FROM AUTHOR]

This paper introduces type P web supercategories. They are defined as diagrammatic monoidal ${\mathbb {k}}$ -linear supercategories via generators and relations. We study the structure of these categories and provide diagrammatic bases for their morphism spaces. We also prove these supercategories provide combinatorial models for the monoidal supercategory generated by the symmetric powers of the natural module and their duals for the Lie superalgebra of type P. [ABSTRACT FROM AUTHOR]

For a finite dimensional representation V of a finite reflection group W , we consider the rational Cherednik algebra H t , c (V , W) associated with (V , W) at the parameters t ≠ 0 and c. The Dunkl total angular momentum algebra O t , c (V , W) arises as the centraliser algebra of the Lie superalgebra osp (1 | 2) containing a Dunkl deformation of the Dirac operator, inside the tensor product of H t , c (V , W) and the Clifford algebra generated by V. We show that when dim ⁡ V ≥ 3 and for every value of the parameter c , the centre of O t , c (V , W) is isomorphic to a univariate polynomial ring. Notably, the generator of the centre changes depending on whether or not (− 1) V is an element of the group W. Using this description of the centre, and using the projection of the pseudo scalar from the Clifford algebra into O t , c (V , W) , we establish results analogous to "Vogan's conjecture" for a family of operators depending on suitable elements of the double cover W ˜. [ABSTRACT FROM AUTHOR]

In this paper, we classify the simple Harish-Chandra modules over the superconformal current algebra g ˆ , which is the semi-direct sum of the N = 1 superconformal algebra with the affine Lie superalgebra g ˙ ⊗ A ⊕ C C 1 , where g ˙ is a finite-dimensional simple Lie algebra, and A is the tensor product of the Laurent polynomial algebra and the Grassmann algebra. As an application, we can directly get the classification of the simple Harish-Chandra modules over the N = 1 Heisenberg-Virasoro algebra. [ABSTRACT FROM AUTHOR]

We construct and investigate certain (unbalanced) superalgebra structures on End K (V) , with K a field of characteristic 0 and V a finite dimensional K -vector space (of dimension n ≥ 2). These structures are induced by a choice of non-degenerate symmetric bilinear form B on V and a choice of non-zero base vector w ∈ V. After exploring the construction further, we apply our results to certain questions concerning integer matrix factorization and isometry of integral lattices. [ABSTRACT FROM AUTHOR]

We find universal characteristic identities for the -split Casimir operator in the representation of the and Lie superalgebras. Using these identities, we construct projectors onto the invariant subspaces of these representations and find universal formulas for their superdimensions. All the formulas are in accordance with the universal description of subrepresentations of the representation of simple basic Lie superalgebras in terms of the Vogel parameters. [ABSTRACT FROM AUTHOR]

In this paper we study extensions between finite-dimensional simple modules over classical Lie superalgebras gl (m | n) , osp (M | 2 n) and q m. We consider a simplified version of the extension graph which is produced from the Ext 1 -graph by identifying representations obtained by parity change and removal of the loops. We give a necessary condition for a pair of vertices to be connected and show that this condition is sufficient in most of the cases. This condition implies that the image of a finite-dimensional simple module under the Duflo-Serganova functor has indecomposable isotypical components. This yields semisimplicity of Duflo-Serganova functor for F in (gl (m | n)) and for F in (osp (M | 2 n)). [ABSTRACT FROM AUTHOR]

We study the representation theory of the Lie superalgebra g l (1 | 1) , constructing two spectral sequences which eventually annihilate precisely the superdimension 0 indecomposable modules in the finite-dimensional category. The pages of these spectral sequences, along with their limits, define symmetric monoidal functors on Rep g l (1 | 1). These two spectral sequences are related by contragredient duality, and from their limits we construct explicit semisimplification functors, which we explicitly prove are isomorphic up to a twist. We use these tools to prove branching results for the restriction of simple modules over Kac-Moody and queer Lie superalgebras to g l (1 | 1) -subalgebras. [ABSTRACT FROM AUTHOR]

We give a new presentation of the Yangian for the orthosymplectic Lie superalgebra osp 1 | 2 m. It relies on the Gauss decomposition of the generator matrix in the R -matrix presentation. The defining relations between the Gaussian generators are derived from a new version of the Drinfeld-type presentation of the Yangian for osp 1 | 2 and some additional relations in the Yangian for osp 1 | 4 by an application of the embedding theorem for the super-Yangians. [ABSTRACT FROM AUTHOR]

We show that there exists a unique crystal base of a parabolic Verma module over a quantum orthosymplectic superalgebra, which is induced from a q -analogue of a polynomial representation of a general linear Lie superalgebra. [ABSTRACT FROM AUTHOR]

We prove that every simple finite dimensional binary Lie superalgebra over the complex numbers field C with non-zero odd part is either a Lie superalgebra or has a solvable even part. [ABSTRACT FROM AUTHOR]

We exhibit a class C of finite dimensional algebras with superinvolution over an algebraically closed field of characteristic zero, with the remarkable property that each member of C generates a minimal variety of algebras with superinvolution. This sums up to the fact that any affine minimal variety of algebras with superinvolution is generated by a suitable member of C , thus providing a complete characterization of the affine minimal varieties of algebras with superinvolution. [ABSTRACT FROM AUTHOR]

We construct Fock and MacMahon modules for the quantum toroidal superalgebra ℰ s associated with the Lie superalgebra m | n and parity s. The bases of the Fock and MacMahon modules are labeled by super-analogs of partitions and plane partitions with various boundary conditions, while the action of generators of ℰ s is given by Pieri-type formulas. We study the corresponding characters. [ABSTRACT FROM AUTHOR]

In this paper we consider the mixed tensor space of a Z 2 -graded vector space. We obtain a spanning set of invariants of the associated symmetric algebra under the action of the general linear supergroup as well as the queer supergroup over the Grassmann algebra. As a consequence, we give a generating set of polynomial invariants for the simultaneous adjoint action of the general linear supergroup on several copies of its Lie superalgebra. We show that in this special case, this set turns out to be the set of supertrace monomials, which is analogous to the result of Procesi for the general linear group. A queer supergroup analogue of these results is also obtained. [ABSTRACT FROM AUTHOR]

In this paper, we explore the super W -algebra SW (3 2 , 3 2) of Neveu-Schwarz type denoted by L , which serves as a supersymmetric conformal algebra. Our investigation focuses on the analysis of simple smooth modules including the Whittaker modules and the highest weight modules over L. Specifically, we utilize simple modules from finite-dimensional solvable Lie superalgebras to construct a multitude of simple smooth modules over L. Additionally, we provide several equivalent descriptions for simple smooth modules over L. [ABSTRACT FROM AUTHOR]

In this paper, we consider the orthogonal symplectic Lie superalgebra $ \mathrm{osp}(1, 2) $ over an algebraically closed field of prime characteristic $ p > 2 $. Using the classification of the simple modules of the Lie superalgebra $ \mathrm{osp}(1, 2) $, we prove that every local superderivation of $ \mathrm{osp}(1, 2) $ to any simple module is a superderivation.

This document provides a comprehensive review of the classification and structure of finite-dimensional simple right-alternative superalgebras. It explores the application of superalgebras in ring theory and the construction of a meaningful structural theory. The document covers various types of superalgebras, including associative superalgebras, commutative alternative superalgebras, and Abelian superalgebras, providing definitions, examples, and classifications. It also discusses the properties and decomposition of different types of superalgebras based on their even parts, as well as the structure of singular superalgebras. The document includes mathematical proofs, examples, and open questions for further research. [Extracted from the article]

The aim of this paper is to transfer the restrictedness theory to Hom-Lie algebras. The concept of restricted Hom-Lie algebras, which is introduced in [S. Bouarroudj and A. Makhlouf, Hom-lie superalgebras in characteristic 2, Mathematics 11 2023, 24, Paper No. 4955], will be used in this paper. First, the existence and uniqueness of p-structures on a Hom-Lie algebra is studied. Then the definition of a restrictable Hom-Lie algebra is given and the equivalence relation between restrictable Hom-Lie algebras and restricted Hom-Lie algebras is constructed. Finally, the p-envelopes of a Hom-Lie algebra are defined and studied. [ABSTRACT FROM AUTHOR]

The main purpose of this paper is to study a generalization of pre-Malcev algebras and pre-Lie superalgebras, which is called pre-Malcev superalgebras. We first introduce the definition of pre-Malcev superalgebras with representations, and obtain some important properties and constructions of the super Kupershmidt operators on pre-Malcev superalgebras. Then we give the relationship among pre-Malcev superalgebras, alternative superalgebras and M-dendriform superalgebras. Furthermore, we give infinitesimal deformations of super Kupershmidt operators. Finally, we define bimodules of pre-Malcev superalgebras and study their relationships with super Kupershmidt operators. [ABSTRACT FROM AUTHOR]

In this paper, we define a partially capable Lie superalgebra. As an application, we classify all capable nilpotent Lie superalgebras of dimension less than or equal to five. [ABSTRACT FROM AUTHOR]

We characterize the finite codimension sub- ${\mathbf {k}}$ -algebras of ${\mathbf {k}}[\![t]\!]$ as the solutions of a computable finite family of higher differential operators. For this end, we establish a duality between such a sub-algebras and the finite codimension ${\mathbf {k}}$ -vector spaces of ${\mathbf {k}}[u]$ , this ring acts on ${\mathbf {k}}[\![t]\!]$ by differentiation. [ABSTRACT FROM AUTHOR]

We prove a conjectural formula for the Brumer–Stark units. Dasgupta and Kakde have shown the formula is correct up to a bounded root of unity. In this paper, we resolve the ambiguity in their result. We also remove an assumption from Dasgupta–Kakde's result on the formula. [ABSTRACT FROM AUTHOR]

Let $H^\infty $ be the algebra of bounded holomorphic functions on the open unit disk, and let $\mathfrak M$ be its maximal ideal space. Let $\mathfrak M_a$ be the union of nontrivial Gleason parts (analytic disks) of $\mathfrak M$. In this paper, we study the problem of extensions of bounded Banach-valued holomorphic functions and holomorphic maps with values in Oka manifolds from Gleason parts of $\mathfrak M_a\setminus \mathbb {D}$. The resulting extensions satisfy the uniform boundedness principle in the sense that their norms are bounded by constants that do not depend on the choice of the Gleason part. The results extend fundamental results of D. Suárez on the characterization of the algebra of restrictions of Gelfand transforms of functions in $H^\infty $ to Gleason parts of $ \mathfrak M_a\setminus \mathbb {D}$. The proofs utilize our recent advances on $\bar \partial $ -equations on quasi-interpolating sets and Runge-type approximations. [ABSTRACT FROM AUTHOR]

In this paper, we present the results related to a problem posed by Andrzej Schinzel. Does the number $N_1(n)$ of integer solutions of the equation $$ \begin{align*}x_1+x_2+\cdots+x_n=x_1x_2\cdot\ldots\cdot x_n,\,\,x_1\ge x_2\ge\cdots\ge x_n\ge 1\end{align*} $$ tend to infinity with n ? Let a be a positive integer. We give a lower bound on the number of integer solutions, $N_a(n)$ , to the equation $$ \begin{align*}x_1+x_2+\cdots+x_n=ax_1x_2\cdot\ldots\cdot x_n,\,\, x_1\ge x_2\ge\cdots\ge x_n\ge 1.\end{align*} $$ We show that if $N_2(n)=1$ , then the number $2n-3$ is prime. The average behavior of $N_2(n)$ is studied. We prove that the set $\{n:N_2(n)\le k,\,n\ge 2\}$ has zero natural density. [ABSTRACT FROM AUTHOR]

Let $\{b_n\}_{n=1}^{\infty }$ be a sequence of integers larger than 1. We will study the harmonic analysis of the equal-weighted Moran measures $\mu _{\{b_n\},\{{\mathcal D}_n\}}$ with ${\mathcal D}_n=\{0,1,2,\ldots ,q_n-1\}$ , where $q_n$ divides $b_n$ for all $n\geq 1.$ In this paper, we first characterize all the maximal orthogonal sets of $L^2(\mu _{\{b_n\},\{{\mathcal D}_n\}})$ via a tree mapping. By this characterization, we give some sufficient conditions for the maximal orthogonal set to be an orthonormal basis. [ABSTRACT FROM AUTHOR]

Suppose $\mathfrak {g}=\mathfrak {g}_{\bar 0}+\mathfrak {g}_{\bar 1}$ is a finite-dimensional restricted Lie superalgebra over an algebraically closed field $\mathbf {k}$ of characteristic $p>2$. In this article, we propose a conjecture for maximal dimensions of irreducible modules over the universal enveloping algebra $U(\mathfrak {g})$ of $\mathfrak {g}$ , as a super generalization of the celebrated first Kac–Weisfeiler conjecture. It is demonstrated that the conjecture holds for all basic classical Lie superalgebras and all completely solvable restricted Lie superalgebras. In this process, we investigate irreducible representations of solvable Lie superalgebras. [ABSTRACT FROM AUTHOR]

In this paper, we prove a rigidity result for the product metric on the product of spheres $S^1\times S^{n-1}$. [ABSTRACT FROM AUTHOR]

In the paper we give some basic properties of the superderivations of Lie superalgebras. Under certain condition, for solvable Lie superalgebras with given nilradicals, we give estimates for upper bounds to the dimensions of complementary subspaces to the nilradicals. Moreover, under the condition that an analogue of Lie's theorem is true, the description of solvable Lie superalgebras of maximal rank is obtained. Namely, we prove that an arbitrary solvable Lie superalgebra of maximal rank under the mentioned condition is isomorphic to the maximal solvable extension of nilradical of maximal rank. Finally, along with effective method of construction of solvable Lie superalgebras of maximal rank we present the description of special type of maximal solvable extension of nilpotent Lie superalgebras. [ABSTRACT FROM AUTHOR]

Exceptional orthogonal Hermite polynomials have been linked to the k-step extension of the harmonic oscillator. The exceptional polynomials allow the existence of different supercharges in the Darboux–Crum and Krein–Adler constructions. They also allow the existence of different types of ladder relations and their associated recurrence relations. The existence of such relations is a unique property of these polynomials. Those relations have been used to construct 2D models which are superintegrable and display an interesting spectrum, degeneracies and finite-dimensional unitary representations. In previous works, only the physical or polynomial part of the spectrum was discussed. It is known that the general solutions are associated with other types of recurrence/ladder relations. We discuss in detail the case of the exceptional Hermite polynomials X 2 (1) and explicitly present new chains obtained by acting with different types of ladder operators. We exploit a recent result by one of the authors (Chalifour and Grundland in Ann Henri Poincaré 21:3341, 2020), where the general analytic solution was constructed and connected with the non-degenerate confluent Heun equation. The analogue Rodrigues formulas for the general solution are constructed. The set of finite states from which the other states can be obtained algebraically is not unique, but the vanishing arrow and diagonal arrow from the diagram of the 2-chain representations can be used to obtain minimal sets. These Rodrigues formulas are then exploited, not only to construct the states, polynomial and non-polynomial, in a purely algebraic way, but also to obtain coefficients from the action of ladder operators in an algebraic manner based on further commutation relations between monomials in the generators. [ABSTRACT FROM AUTHOR]

AbstractIn this paper, we extend the notion of anti-pre-Lie algebras to the ℤ2-graded version, and introduce the notion of anti-pre-Lie superalgebras. They can be characterized as a class of Lie-admissible superalgebras that satisfy its negative left multiplication operators are the representations of the corresponding sub-adjacent Lie superalgebras. And we give the classification of 2-dimensional anti-pre-Lie superalgebras. We introduce the notion of anti-super -operators on Lie superalgebras to explore the relationships between anti-super -operators and anti-pre-Lie superalgebras. We show that nondegenerate super-commutative 2-cocycles on Lie superalgebras can obtain a class of compatible anti-pre-Lie superalgebra structures. In addition, we introduce a subclass of anti-pre-Lie superalgebras, namely admissible Novikov superalgebras, which correspond to Novikov superalgebras through q-superalgebras. Finally, we introduce the notions of anti-pre-Lie Poisson superalgebras and admissible Novikov-Poisson superalgebras, extending the correspondence to the level of Poisson type structures, the correspondence of the Novikov-Poisson superalgebras and the admissible Novikov-Poisson superalgebras are realized through admissible pairs. [ABSTRACT FROM AUTHOR]

A twisted affine Lie superalgebra is either a twisted affine Lie algebra or of one of the types X = A (2 m − 1 , 2 n − 1) (2) ( m , n ≠ 0 , (m , n) ≠ (1 , 1) ), A (2 m , 2 n) (4) , A (2 m , 2 n − 1) (2) or D (m + 1 , n) (2) ( m ≥ 0 , n > 0 ). It is known that irreducible integrable highest weight modules over a twisted affine Lie superalgebra of type X do not exist if m ≠ 0. In this paper, we show that nonzero level irreducible integrable finite weight modules over a twisted affine Lie superalgebra of type X do not exist if m ≠ 0. [ABSTRACT FROM AUTHOR]

In this paper, we consider the orthogonal symplectic Lie superalgebra osp(1, 2) over an algebraically closed field of prime characteristic p > 2. Using the classification of the simple modules of the Lie superalgebra osp(1, 2), we prove that every local superderivation of osp(1, 2) to any simple module is a superderivation. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to study some results of constructions on Hom–Poisson superalgebras. We use the representations and Rota–Baxter operators. We introduce the notion of n -ary Hom–Nambu–Poisson superalgebras and their representations and we study relationships between Hom–Poisson superalgebras and its induced n -ary Hom–Nambu–Poisson superalgebras. The same concept is applied for the representations of n -ary Hom–Nambu–Poisson superalgebras. [ABSTRACT FROM AUTHOR]

AbstractIn this paper, we characterize a relative Rota-Baxter operator on an n-Lie superalgebra with a representation (called an n-LSRep pair) as a Maurer-Cartan element of a certain Lie n-algebra. Then, we introduce the notions of n-pre-Lie superalgebras and symplectic n-Lie superalgebras and give some related results based on relative Rota-Baxter operators. In the next part, we give the cohomologies of relative Rota-Baxter operators on n-LSRep pairs and study infinitesimal deformations and extensions of finite order deformations of relative Rota-Baxter operators through the cohomology groups of relative Rota-Baxter operators. Finally, we build the relationship between the cohomology groups of relative Rota-Baxter operators on n-LSRep pairs and those on (n+1)-LSRep pairs by certain linear functions. [ABSTRACT FROM AUTHOR]

AbstractOver a field of characteristic p > 2, firstly, the structure of Kac modules of Lie superalgebra P˜(2) and the weight space decompositions are given. Secondly, the weight-derivations of P˜(2) to its Kac modules are computed. Finally, the first cohomology of P˜(2) with coefficients in Kac modules is determined. [ABSTRACT FROM AUTHOR]

The goal of this paper is to generalize a statement by Drinfeld, asserting that Yangians can be constructed as limit forms of the quantum loop algebras, to the super case. We establish a connection between quantum loop superalgebra and super Yangian of the general linear Lie superalgebra gl M | N in RTT type presentation. In particular, we derive the Poincaré-Birkhoff-Witt(PBW) theorem for the quantum loop superalgebra U q (Lgl M | N). Additionally, we investigate the application of the same argument to twisted super Yangian of the ortho-symplectic Lie superalgebra. For this purpose, we introduce the twisted quantum loop superalgebra as a one-sided coideal of U q (Lgl M | 2 n) with respect to the comultiplication. [ABSTRACT FROM AUTHOR]

AbstractWe explore two modules F and K over a subalgebra U of the quantum group Uq(gl(∞)). The first U-module is the Fock space related to the natural module over the quantum group. If we let O+ to be the category of finite-dimensional osp(2|2)-modules of integral weights, the second U-module K is the vector space obtained by tensoring the Grothendieck group K(O+) by the field Q. Using thick tensor ideals and a U-isomorphism, we compute the socles of F and K. [ABSTRACT FROM AUTHOR]

Let F be an algebraically closed field of characteristic zero. We give a complete classification of finite dimensional simple superalgebras over F endowed with a graded automorphism or a pseudoautomorphism. [ABSTRACT FROM AUTHOR]

We describe a Yang-Baxter integrable vertex model, which can be realized as a degeneration of a vertex model introduced by Aggarwal, Borodin, and Wheeler (2021). From this vertex model, we construct a certain class of partition functions that we show are essentially equal to the super ribbon functions of Lam. Using the vertex model formalism, we give proofs of many properties of these polynomials, namely a Cauchy identity and generalizations of known identities for supersymmetric Schur polynomials. [ABSTRACT FROM AUTHOR]

The Lie superalgebra gl(m|n) admits irreducible, finite-dimensional representations that continuously depend on a free parameter. For each representation, we define a generalised Gaudin superalgebra through an associated solution of the classical Yang-Baxter equation. The universal enveloping superalgebra of the Gaudin superalgebra formally contains a commuting family of transfer matrices, given by a universal expression. It will be shown that in many instances these transfer matrices are identically zero in the universal enveloping superalgebra. We offer an alternative formulation for identifying transfer matrices.

In this paper, we classify the gradings up to equivalence on the filiform Lie superalgebras Lm,n over an algebraically closed field of characteristic 0. Moreover, by the relation between a grading and a standard grading, we describe all pairwise inequivalent gradings and their universal groups of Lm,n. [ABSTRACT FROM AUTHOR]

Let 픤 = 픤 0 ¯ ⊕ 픤 1 ¯ {{\mathfrak{g}}={\mathfrak{g}}_{\bar{0}}\oplus{\mathfrak{g}}_{\bar{1}}} be a basic classical Lie superalgebra over an algebraically closed field 퐤 {{\mathbf{k}}} of characteristic p > 2 {p>2} . Denote by 풵 {\mathcal{Z}} the center of the universal enveloping algebra U ⁢ ( 픤 ) {U({\mathfrak{g}})} . Then 풵 {\mathcal{Z}} turns out to be finitely-generated purely-even commutative algebra without nonzero divisors. In this paper, we demonstrate that the fraction Frac ⁡ ( 풵 ) {\operatorname{Frac}(\mathcal{Z})} is isomorphic to Frac ⁡ ( ℨ ) {\operatorname{Frac}(\mathfrak{Z})} for the center ℨ {\mathfrak{Z}} of U ⁢ ( 픤 0 ¯ ) {U({\mathfrak{g}}_{\bar{0}})} . Consequently, both Zassenhaus varieties for 픤 {{\mathfrak{g}}} and 픤 0 ¯ {{\mathfrak{g}}_{\bar{0}}} are birationally equivalent via a subalgebra 풵 ~ ⊂ 풵 {\widetilde{\mathcal{Z}}\subset\mathcal{Z}} , and Spec ⁡ ( 풵 ) {\operatorname{Spec}(\mathcal{Z})} is rational under the standard hypotheses. [ABSTRACT FROM AUTHOR]

The natural filtrations of infinite-dimensional Lie superalgebras of O -type over a field of positive characteristic are proved to be invariant under automorphisms by studying ad-nilpotent elements. We observe that two Lie superalgebras of O -type are isomorphic if and only if the integers in the definition of these Lie superalgebras are intrinsic. Thereby, all the Lie superalgebras of O -type are classified in the sense of isomorphism. [ABSTRACT FROM AUTHOR]

We develop and study the generalization of rational Schur algebras to the super setting. Similar to the classical case, this provides a new method for studying rational supermodules of the general linear supergroup GL (m | n). Furthermore, we establish a Schur-Weyl duality result for rational Schur superalgebras and conclude that under certain conditions these objects will be semisimple. [ABSTRACT FROM AUTHOR]

Based on Lie superalgebra sI(1, 2) and the TAH scheme, we derive (1+1)-dimensional and (2+1)-dimensional nonisospectral integrable hierarchies and the corresponding super Hamiltonian structures. At the same time, we construct a generalized Lie superalgebra sI(1, 2), and apply it to (1+1)-dimensional and (2+1)-dimensional integrable systems. Finally, we discuss the super Hamiltonian structures of (1+1)-dimensional and (2+1)-dimensional integrable hierarchies associated with Lie superalgebra G sI(1, 2). [ABSTRACT FROM AUTHOR]

We introduce the concepts of complex finite-dimensional contact and Frobenius Lie superalgebras, including some relevant examples of this type of Lie superalgebras. We establish some basic properties satisfied by both contact and Frobenius Lie superalgebras, providing characterizations for both of them in terms of its associated structure supermatrix. Finally, if g = g 0 ⊕ g 1 is a homogeneous even Frobenius Lie superalgebra satisfying dim g 0 ≠ dim g 1 , we show that such g contains a contact superideal of codimension one. [ABSTRACT FROM AUTHOR]

This article is the final one in the series of papers by the authors devoted to finite-dimensional singular (simple right-alternative with zero product in the even part) superalgebras. In previous papers, algebraically generated singular superalgebras were studied and it was proved that any such superalgebra has the structure of an extended double. In this article, we prove that any singular superalgebra is algebraically generated and, therefore, has the structure of an extended double. [ABSTRACT FROM AUTHOR]

Lie color algebras generalize Lie superalgebras. We adapt a construction of Bahturin and Pagon to create enhanced Lie color algebras. This construction offers a much-needed method for constructing simple Lie color algebras. To illustrate its applicability, we demonstrate how to enhance any simple Lie superalgebra and obtain a simple Lie color algebra. Additionally, we show that if a Lie color algebra has a nonzero determinant, this property extends to its enhancement. This ensures its universal enveloping algebra is semiprime. Extra conditions on the grading group provide a primeness criterion. [ABSTRACT FROM AUTHOR]

Over a field of characteristic p > 2 , the first cohomology of the Witt-type Lie superalgebra W(2) with coefficients in all Kac modules K (λ) is determined by use of the weight-derivation spaces from W(2) to K (λ) . [ABSTRACT FROM AUTHOR]