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Suppose C is a cyclic Galois cover of the projective line branched at the three points 0, 1, and ∞ . Under a mild condition on the ramification, we determine the structure of the graded Lie algebra of the lower central series of the fundamental group of C in terms of a basis which is well-suited to studying the action of the absolute Galois group of Q . [ABSTRACT FROM AUTHOR]

In Fialowski's classification for algebras of maximal class, there are three Lie algebras of maximal class with one-dimensional homogeneous components: $ \mathfrak {m}_0 $ m 0 , $ L_1 $ L 1 and $ \mathfrak {m}_2 $ m 2 . In this paper, we study their biderivations by considering the embedded mapping to derivation algebras. Then we determine commuting mappings on these algebras as an application of biderivations. Finally, local and 2-local derivations for these three algebras are characterized as the given gradings. [ABSTRACT FROM AUTHOR]

AbstractIn this paper, we use the Balinsky-Novikov construction to naturally realize the super Heisenberg-Virasoro algebra g . We then proceed to compute the second cohomology of g with trivial coefficients and determine its universal central extension. This yields two odd 2-cocycles in addition to the Virasoro 2-cocycle. We also calculate the first cohomology group of g in the adjoint module to classify all derivations of g . Finally, we classify all automorphisms of g and describe its automorphism group. [ABSTRACT FROM AUTHOR]

Given an integer n, we introduce the integral Lie ring of partitions with bounded maximal part, whose elements are in one-to-one correspondence to integer partitions with parts in { 1 , 2 , ⋯ , n - 1 } . Starting from an abelian subring, we recursively define a chain of idealizers and we prove that the sequence of ranks of consecutive terms in the chain is ultimately periodic. Moreover, we show that its growth depends of the partial sum of the partial sum of the sequence counting the number of partitions. This work generalizes our previous recent work on the same topic, devoted to the modular case where partitions were allowed to have a bounded number of repetitions of parts in a ring of coefficients of positive characteristic. [ABSTRACT FROM AUTHOR]

In this paper, the super-skewsymmetric super-biderivations on the N = 1 super Heisenberg-Virasoro algebra are completely determined. Furthermore, we find that every super-skewsymmetric super-biderivation of the N = 1 super Heisenberg-Virasoro algebra is inner. [ABSTRACT FROM AUTHOR]

In this paper, we study the finite-dimensional irreducible representations of the BC-graded Lie algebra g 2 n + 1 (ℂ q) by investigating the structure of the BC-graded Lie algebra g 2 n + 1 (R) , where R is a unital involutory associative algebra over a field F of characteristic zero. [ABSTRACT FROM AUTHOR]

Given two Lie ∞ -algebras E and V, any Lie ∞ -action of E on V defines a Lie ∞ -algebra structure on E ⊕ V . Some compatibility between the action and the Lie ∞ -structure on V is needed to obtain a particular Loday ∞ -algebra, the non-abelian hemisemidirect product. These are the coherent actions. For coherent actions it is possible to define non-abelian homotopy embedding tensors as Maurer-Cartan elements of a convenient Lie ∞ -algebra. Generalizing the classical case, we see that a non-abelian homotopy embedding tensor defines a Loday ∞ -structure on V and is a morphism between this new Loday ∞ -algebra and E. [ABSTRACT FROM AUTHOR]

The compatible left-symmetric superalgebra structures on the super Galilean conformal algebra with some natural grading conditions are completely determined. The results of earlier work on left-symmetric superalgebra structures on the infinite-dimensional Lie superalgebra g studied by Chen and Hassan play an essential role in determining these compatible structures. [ABSTRACT FROM AUTHOR]

In this paper, we'll introduce the concept of sympathetic Lie color algebras and show that some classical properties of semi-simple Lie color algebras are still valid for sympathetic Lie color algebras. We prove that every Lie color algebras L contains a greatest sympathetic ideal M, and there exists a solvable ideal of L denoted by P(L) which is the greatest among the solvable ideals I of L for which I ∩ M = { 0 } . And we show that there exists a sympathetic subalgebra m of L such that L = m ⊕ P (L) and L is a sympathetic Lie color algebra if and only if P (L) = { 0 } . What's more, we also study the ideals I of a Lie color algebra L such that L/I is a sympathetic Lie color algebra and get some properties about them. [ABSTRACT FROM AUTHOR]

Let W ⁢ (n) be the Jacobson–Witt algebra over algebraic closed field 핂 with characteristic p > 2 . In [K. Ou and B. Shu, Borel subalgebras of restricted Cartan-type Lie algebras, J. Algebra Appl. 21 2022, 11, Paper No. 2250210], we introduced the so-called B-subalgebra of W ⁢ (n) , which serves as an analog of the Borel subalgebra of classical Lie algebras. As a sequel, we describe the structure of the variety ℬ consisting of all B-subalgebras of W ⁢ (n) in this paper. This variety presents an analog of the flag variety for classical Lie algebras. It is shown that ℬ is related to the variety of all full flags in 핂 n + 1 . Additionally, we provide a detailed description of the varieties for W ⁢ (1) as an illustrative example. With the above setting-up, one may establish the Springer theory and geometric representations for the Jacobson–Witt algebras. [ABSTRACT FROM AUTHOR]

In this note, we determine the structure of the associative algebra generated by the differential operators μ¯,∂¯,∂\overline{\mu },\overline{\partial },\partial , and μ\mu that act on complex-valued differential forms of almost complex manifolds. This is done by showing that it is the universal enveloping algebra of the graded Lie algebra generated by these operators and determining the structure of the corresponding graded Lie algebra. We then determine the cohomology of this graded Lie algebra with respect to its canonical inner differential [d,−]\left[d,-], as well as its cohomology with respect to all its inner differentials.

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]

Let U T n (F) be the algebra of the n × n upper triangular matrices and denote U T n (F) (-) the Lie algebra on the vector space of U T n (F) with respect to the usual bracket (commutator), over an infinite field F. In this paper, we give a positive answer to the Specht property for the ideal of the Z n -graded identities of U T n (F) (-) with the canonical grading when the characteristic p of F is 0 or is larger than n - 1 . Namely we prove that every ideal of graded identities in the free graded Lie algebra that contains the graded identities of U T n (F) (-) , is finitely based. Moreover we show that if F is an infinite field of characteristic p = 2 then the Z 3 -graded identities of U T 3 (-) (F) do not satisfy the Specht property. More precisely, we construct explicitly an ideal of graded identities containing that of U T 3 (-) (F) , and which is not finitely generated as an ideal of graded identities. [ABSTRACT FROM AUTHOR]

The purpose of this paper is twofold. Firstly, to emphasise that the class of Lie algebras with chain lattices of ideals are elementary blocks in the embedding or decomposition of Lie algebras with finite lattice of ideals. Secondly, to show that the number of Lie algebras of this class is large and they support other types of Lie structures. Beginning with general examples and algebraic decompositions, we focus on computational algorithms to build Lie algebras in which the lattice of ideals is a chain. The chain condition forces gradings on the nilradicals of this class of algebras. Our algorithms yield to several positive naturally graded parametric families of Lie algebras. Further generalizations and other kind of structures will also be discussed. [ABSTRACT FROM AUTHOR]

We compute the graded polynomial identities and its graded codimension sequence for the elementary gradings of the Lie algebra of upper triangular matrices of order 3. [ABSTRACT FROM AUTHOR]

The paper is devoted to introducing graded systems of quotients of graded Lie triple systems, graded weak systems of quotients of graded Lie triple systems and graded systems of Martindale-like quotients of graded Lie triple systems with respect to power graded filters of sturdy graded ideals. And then we investigate certain vital properties of graded Lie triple systems, which can be lifted from a graded Lie triple system to its graded systems of quotients. Furthermore, the connections among graded systems of quotients, graded weak systems of quotients and graded systems of Martindale-like quotients are surveyed. What's more, the relationship between graded systems of quotients of graded Lie triple systems and systems of quotients of Lie triple systems is highlighted. Besides, the relationship between graded systems of quotients of a graded Lie triple system and graded algebras of quotients of the graded Lie algebra L(T) (the standard embedding of a graded Lie triple system T) is considered, too. [ABSTRACT FROM AUTHOR]

Complete hom-Lie superalgebras are considered and some equivalent conditions for a hom-Lie superalgebra to be a complete hom-Lie superalgebra are established. In particular, the relation between decomposition and completeness for a hom-Lie superalgebra is described. Moreover, some conditions that the linear space of α s -derivations of a hom-Lie superalgebra to be complete and simply complete are obtained. [ABSTRACT FROM AUTHOR]

The purpose is to introduce the notions of 3-Bihom-ρ-Lie algebras and 3-pre-Bihom-ρ-Lie algebras. The authors describe their constructions and express the related lemmas and theorems. Also, they define the 3-Bihom-ρ-Leibniz algebras and show that a 3-Bihom-ρ-Lie algebra is a 3-Bihom-ρ-Leibniz algebra with the ρ-Bihom-skew symmetry property. Moreover, a combination of a 3-Bihom-ρ-Lie algebra bracket and a Rota-Baxer operator gives a 3-pre-Bihom-ρ-Lie algebra structure. [ABSTRACT FROM AUTHOR]

Schrödinger-Virasoro Lie algebras spanned by { L n , Y p , M k | n , k ∈ Z , p ∈ 1 2 + Z } are annihilation algebras of Lie conformal algebras of rank three, which are widely studied by mathematicians and physicists. In this article, we introduce the notion of Schrödinger-Virasoro Lie conformal algebras. Using the pseudoalgebra language, we study the Schrödinger-Virasoro Lie conformal algebras and classify all Schrödinger-Virasoro Lie conformal algebras. [ABSTRACT FROM AUTHOR]

Let K be any field of characteristic two and let $U_1$ and $W_1$ be the Lie algebras of the derivations of the algebra of Laurent polynomials $K[t,t^{-1}]$ and of the polynomial ring K [ t ], respectively. The algebras $U_1$ and $W_1$ are equipped with natural $\mathbb{Z}$ -gradings. In this paper, we provide bases for the graded identities of $U_1$ and $W_1$ , and we prove that they do not admit any finite basis. [ABSTRACT FROM AUTHOR]

We study conformal biderivations of a Lie conformal algebra. First, we give the definition of a conformal biderivation. Next, we determine the conformal biderivations of loop W(a, b) Lie conformal algebra, loop Virasoro Lie conformal algebra, and Virasoro Lie conformal algebra. Especially, all conformal biderivations on Virasoro Lie conformal algebra are inner conformal biderivations. [ABSTRACT FROM AUTHOR]

For any decomposition of a Lie superalgebra G into a direct sum G = H ⊕ E of a subalgebra H and a subspace E , without any further resctrictions on H and E , we construct a nonlinear realization of G on E. The result generalizes a theorem by Kantor from Lie algebras to Lie superalgebras. When G is a differential graded Lie algebra, we show that it gives a construction of an associated L ∞ -algebra. [ABSTRACT FROM AUTHOR]

Let g be the Witt algebra over an infinite field of prime characteristic p. In this paper, we discuss the properties of local derivations on g , and prove that every local derivation on g is a derivation. Moreover, it is shown that there exist some local derivations which are not derivations for the maximal subalgebra g 0 of g . [ABSTRACT FROM AUTHOR]

A classification up to automorphism of the inner ideals of the real finite-dimensional simple Lie algebras is given, jointly with precise descriptions in the case of the exceptional Lie algebras. [ABSTRACT FROM AUTHOR]

In this article, we compute biderivations of the q-deformed Heisenberg–Virasoro algebra. As an application, linear commuting maps are characterized. We also provide calculations of α-derivations and α-biderivations of the q-deformed Heisenberg–Virasoro algebra and obtain that it has no nonzero α-derivations and α-biderivations. [ABSTRACT FROM AUTHOR]

We define O -operators on a Lie ∞-algebra E with respect to an action of E on another Lie ∞-algebra and we characterize them as Maurer-Cartan elements of a certain Lie ∞-algebra obtained by Voronov's higher derived brackets construction. The Lie ∞-algebra that controls the deformation of O -operators with respect to a fixed action is determined. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to study the relationships between a BiHom-Lie superalgebras and its induced 3-BiHom-Lie superalgebras. We introduce the notion of (α s , β r) -derivation, (α s , β r) -quasiderivation and generalized (α s , β r) -derivation of 3-BiHom-Lie superalgebras and their relation with derivation of BiHom-Lie superalgebras. We introduce also the concepts of Rota-Baxter operators and Nijenhuis Operators of 3-BiHom-Lie superalgebras. We also explore the construction of 3-BiHom-Lie superalgebras by using Rota-Baxter of BiHom-Lie superalgebras. [ABSTRACT FROM AUTHOR]

Let 𝔽 be an arbitrary field of characteristic p > 2. In this paper we study irreducible modules with highest weight vectors over Witt and special Lie superalgebras of 𝔽. The same irreducible modules of general and special linear Lie superalgebras, which are the 0-th part of Witt and special Lie superalgebras in certain ℤ-grading, are also considered. Then we establish a certain connection called a P-expansion between these modules.

We classify all integrable triples in simple Lie algebras, up to equivalence. The importance of this problem stems from the fact that for each such equivalence class one can construct the corresponding integrable hierarchy of bi-Hamiltonian PDE. The simplest integrable triple (f, 0, e) in sl 2 corresponds to the KdV hierarchy, and the triple (f , 0 , e θ) , where f is the sum of negative simple root vectors and e θ is the highest root vector of a simple Lie algebra, corresponds to the Drinfeld–Sokolov hierarchy. [ABSTRACT FROM AUTHOR]

We generalize the P(N)-graded Lie superalgebras of Martinez-Zelmanov. This generalization is not so restrictive but sufficient enough so that we are able to have a classification for this generalized P(N)-graded Lie superalgebras. Our result is that the generalized P(N)-graded Lie superalgebra L is centrally isogenous to a matrix Lie superalgebra coordinated by an associative superalgebra with a super-involution. Moreover, L is P(N)-graded if and only if the coordinate algebra R is commutative and the super-involution is trivial. This recovers Martinez-Zelmanov's theorem for type P(N). We also obtain a generalization of Kac's coordinatization via Tits-Kantor-Koecher construction. Actually, the motivation of this generalization comes from the Fermionic-Bosonic module construction. [ABSTRACT FROM AUTHOR]

This paper aims to research some basic results on complete Bihom-Lie superalgebras. Particularly, we give some sufficient conditions for a Bihom-Lie superalgebra to be complete. Afterwards, we determine explicitly the derivation superalgebras of centerless perfect Bihom-Lie superalgebras of arbitrary dimension and characteristic over any field are complete. Meanwhile some related properties of perfect Bihom-Lie superalgebras are obtained by the concept of the perfect ideal. [ABSTRACT FROM AUTHOR]