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In this paper, first we give the notion of a compatible 3-Lie algebra and construct a bidifferential graded Lie algebra whose Maurer-Cartan elements are compatible 3-Lie algebras. We also obtain the bidifferential graded Lie algebra that governs deformations of a compatible 3-Lie algebra. Then we introduce a cohomology theory of a compatible 3-Lie algebra with coefficients in itself and show that there is a one-to-one correspondence between equivalent classes of infinitesimal deformations of a compatible 3-Lie algebra and the second cohomology group. We further study 2-order 1-parameter deformations of a compatible 3-Lie algebra and introduce the notion of a Nijenhuis operator on a compatible 3-Lie algebra, which could give rise to a trivial deformation. At last, we introduce a cohomology theory of a compatible 3-Lie algebra with coefficients in arbitrary representation and classify abelian extensions of a compatible 3-Lie algebra using the second cohomology group. [ABSTRACT FROM AUTHOR]

In this paper, we construct a new integrable hierarchy by using charged free boson and Friedan–Martinec–Shenker (FMS) bosonization, called modified bosonic integrable hierarchy, which is an anologue of the classical (l − l ′) –th modified KP hierarchy. Firstly, the representation of g l ˆ ∞ is given by using charged free bosons. Then the modified bosonic integrable hierarchy is proposed. Next after discussing FMS bosonizations in the language of lattice vertex algebra, bilinear equations of modified bosonic integrable hierarchy are constructed by FMS bosonization, which are rewritten into Hirota bilinear equations by using Hirota derivatives. At last, we consider solutions of this modified bosonic integrable hierarchy. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the concepts of relative and absolute Ω-Rota-Baxter algebras of weight λ , which can be considered as a family algebraic generalization of relative and absolute Rota-Baxter algebras of weight λ. We study the deformations of relative and absolute Ω-Rota-Baxter algebras of arbitrary weight. Explicitly, we construct an L ∞ [ 1 ] -algebra via the method of higher derived brackets, whose Maurer-Cartan elements correspond to relative Ω-Rota-Baxter algebra structures of weight λ. For a relative Ω-Rota-Baxter algebra of weight λ , the corresponding twisted L ∞ [ 1 ] -algebra controls its deformations, which leads to the cohomology theory of it, and this cohomology theory can interpret the formal deformations of the relative Ω-Rota-Baxter algebra. Moreover, we also obtain the corresponding results for absolute Ω-Rota-Baxter algebras of weight λ from the relative version. [ABSTRACT FROM AUTHOR]

In this paper, we study non-weight representations of another two classes of Lie superalgebras T (p , s) of Block type, where p ∈ C and s = 0 , 1. We completely classify the free U (h) -modules of rank (1 + 1) over T (p , s). As byproducts, we also obtain the non-weight representation results for four classes of non-semisimple finitely graded super-extensions of the Virasoro algebra. [ABSTRACT FROM AUTHOR]

In this paper, we present a framework to construct sequences of spectral triples on top of an inductive sequence defining an AF -algebra. One aim of this paper is to lift arrows of a Bratteli diagram to arrows between Krajewski diagrams. The spectral actions defining Non-commutative Gauge Field Theories associated to two spectral triples related by these arrows are compared (tensored by a commutative spectral triple to put us in the context of Almost Commutative manifolds). This paper is a follow up of a previous one in which this program was defined and physically illustrated in the framework of the derivation-based differential calculus, but the present paper focuses more on the mathematical structure without trying to study the physical implications. [ABSTRACT FROM AUTHOR]

In [16] , we studied deformations A α , β of the second Weyl algebra and computed their derivations. In the present paper, we identify the semiclassical limits A α , β of these deformations and compute their Poisson derivations. Our results show that the first Hochschild cohomology group HH 1 (A α , β) is isomorphic to the first Poisson cohomology group HP 1 (A α , β). [ABSTRACT FROM AUTHOR]

Let d be a positive integer. The Yangian Y d = Y (gl (d , C)) of the general linear Lie algebra gl (d , C) has countably many generators and quadratic-linear defining relations, which can be packed into a single matrix relation using the Yang matrix — the famous RTT presentation. Alternatively, Y d can be built from certain centralizer subalgebras of the universal enveloping algebras U (gl (N , C)) , with the use of a limit transition as N → ∞. This approach is called the centralizer construction. The paper shows that a generalization of the centralizer construction leads to a new family { Y d , L : L = 1 , 2 , 3 , ... } of Yangian-type algebras (the Yangian Y d being the first term of this family). For the new algebras, the RTT presentation seems to be missing. Nevertheless, the algebras Y d , L share a number of properties of the Yangian Y d , including the existence of defining quadratic-linear commutation relations. [ABSTRACT FROM AUTHOR]

In this paper, we study Nijenhuis operators on Leibniz algebras. We discuss the relationship of Nijenhuis operators with Rota-Baxter operators and modified Rota-Baxter operators on Leibniz algebras. We define a representation theory of Nijenhuis Leibniz algebras and construct a cohomology theory. Next, we define a one-parameter formal deformation theory of Nijenhuis Leibniz algebras and study infinitesimals, rigidity, and equivalences along the line of Gerstenhaber deformation theory. As applications of our cohomology theory, we show that our cohomology is deformation cohomology and study abelian extensions of such algebras. [ABSTRACT FROM AUTHOR]

In this paper, we first study the twisted pre-Lie algebra of connected finite topological spaces. Then we construct the twisted pre-Lie structure on its doubling space. We describe the enveloping algebra of both twisted pre-Lie algebras, and we prove that the doubling space of connected topological spaces is a left module on the twisted pre-Lie algebra. We exhibit natural semi-products of both twisted pre-Lie algebras and both enveloping algebras respectively. [ABSTRACT FROM AUTHOR]

The stable envelopes of Okounkov et al. realize some representations of quantum algebras associated to quivers, using geometry. We relate these geometric considerations to quantum field theory. The main ingredients are the supersymmetric interfaces in gauge theories with four supercharges, relation of supersymmetric vacua to generalized cohomology theories, and Berry connections. We mainly consider softly broken compactified three dimensional N = 4 theories. The companion papers will discuss applications of this construction to symplectic duality, Bethe/gauge correspondence, generalizations to higher dimensional theories, and other topics. [ABSTRACT FROM AUTHOR]

In this paper, we study the non-weight modules over the Block type algebra B ′ (p , q) , where p (≠ 0) , q are some complex numbers. We construct a family of such irreducible modules, determine their isomorphism classes, and show that they exhaust the B ′ (p , q) -modules which are U (h) -free modules of rank 1, where h is the Cartan subalgebra. [ABSTRACT FROM AUTHOR]

Algebraic structures relative to a semigroup Ω (also called family algebraic structures) first appeared in the work of renormalization in quantum field theory. In this paper, we first consider Ω-diassociative algebras and diassociative family algebras as family analogues of diassociative algebras. We define the cohomologies of these algebras generalizing the cohomology of diassociative algebras introduced by Frabetti. We also introduce (relative) averaging family operators as the family analogue of averaging operators and show that they induce diassociative family algebras and Ω-diassociative algebras. Next, we define the cohomology of a (relative) averaging family operator as the cohomology of the induced Ω-diassociative algebra with coefficients in a suitable representation. Finally, as an application of our cohomology, we study formal one-parameter deformations of relative averaging family operators. [ABSTRACT FROM AUTHOR]

In this paper, we consider compatible Hom-Lie algebras as a twisted version of compatible Lie algebras. Compatible Hom-Lie algebras are characterized as Maurer-Cartan elements in a suitable bidifferential graded Lie algebra. We also define a cohomology theory for compatible Hom-Lie algebras generalizing the recent work of Liu, Sheng and Bai. As applications of cohomology, we study abelian extensions and deformations of compatible Hom-Lie algebras. [ABSTRACT FROM AUTHOR]

Zusmanovich gave a fundamental result on nontrivial ω -Lie algebras. But the structure of ω -Lie algebras is still unclear. In this paper, we give a theorem on the structure of nontrivial ω -Lie algebras over C. [ABSTRACT FROM AUTHOR]

In this paper we define an equivariant Floer A ∞ algebra for C and C P 1 by using Cartan model. We then prove an equivariant homological mirror symmetry, i.e. an equivalence between an A ∞ category of equivariant Lagrangian branes and the category of matrix factorizations of Givental's equivariant Landau-Ginzburg potential function. [ABSTRACT FROM AUTHOR]

In this paper, we classify modules which are free of rank 1 over the negative part for the Schrödinger algebra S = sl 2 ⋉ H. Sufficient and necessary conditions for such modules to be simple are given. From these modules, we construct examples of simple S -modules which are free of arbitrary rank over U (C h). [ABSTRACT FROM AUTHOR]

The notion of a Manin triple of Lie algebras admits a generalization, to dg Lie algebras, in which various properties are required to hold only up to homotopy. This paper introduces two classes of examples of such homotopy Manin triples. These examples are associated to analogs in complex dimension two of, respectively, the punctured formal 1-disc, and the complex plane with multiple punctures. The dg Lie algebras which appear include certain higher current algebras in the sense of Faonte, Hennion and Kapranov [18]. We work in a ringed space we call rectilinear space , and one of the tools we introduce is a model of the derived sections of its structure sheaf, whose construction is in the spirit of the adelic complexes for schemes due to Parshin and Beilinson. [ABSTRACT FROM AUTHOR]

In this paper, first we discuss Hom-pre-Poisson algebras and their relationships with Hom-Poisson algebra. Then we introduce the notion of a Hom-pre-Gerstenhaber algebra and show that a Hom-pre-Gerstenhaber algebra gives rise to a Hom-Gerstenhaber algebra. Moreover, we consider Hom-dendriform formal deformations of Hom-zinbiel algebras and show that Hom-pre-Poisson algebras are the corresponding semi-classical limits. Furthermore, we consider Hom- O -operators on Hom-Poisson algebras and study their relationships with Hom-pre-Poisson algebras. Finally, we define the notion of dual-Hom-pre-Poisson algebra and show that a Hom-average-operator on a Hom-Poisson algebra naturally gives rise to a dual-Hom-pre-Poisson algebra. [ABSTRACT FROM AUTHOR]

In the present paper, we define a quantum analog of Lotka-Volterra algebras using a coalgebra scheme. This new framework provides a fresh perspective for the treatment of generic algebras. Additionally, a flow of quantum analogs of Lotka-Volterra genetic algebras is investigated. It's worth mentioning that such types of algebras are first introduced in this work. We observe that a flow of algebras is a particular type of continuous-time dynamical system, with states that are algebras and a structural constant matrix that depends on time and satisfies certain analogs of the Kolmogorov-Chapman equations. Using quantum quadratic operators, it is constructed a flow of quantum Lotka-Volterra algebras for the given multiplication. Furthermore, time-dependent behavior properties of these flow algebras are examined. The algebraic properties of the introduced flows are also studied, such as finding idempotents and examining an algebra generated by a pair of idempotents. It is shown that the later one is associative, while the flow is not associative. Additionally, derivations of the flow of the algebras are described. [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]

In this paper, we study non-weight representations of two classes of Lie algebras and two classes of Lie superalgebras of Block type. More precisely, we completely classify the free U (h) -modules of rank one or two over these Lie superalgebras. Then we determine the simplicity and isomorphism classes of these modules. As a byproduct, we also obtain the non-weight representation results for a class of non-semisimple finitely graded extensions of the Virasoro algebra. [ABSTRACT FROM AUTHOR]

This paper studies local properties of the Virasoro-like algebra. Namely, the 2-local derivations and 2-local automorphisms of the Virasoro-like algebra are considered. It is proved that every 2-local derivation of this Lie algebra is a derivation, and every 2-local automorphism of this Lie algebra is an automorphism. [ABSTRACT FROM AUTHOR]

A LieYRep pair consists of a Lie-Yamaguti algebra and its representation. In this paper, we establish the cohomology theory of LieYRep pairs and characterize their linear deformations by the second cohomology group. Then we introduce the notion of relative Rota-Baxter-Nijenhuis structures on LieYRep pairs, investigate their properties, and prove that a relative Rota-Baxter-Nijenhuis structure gives rise to a pair of compatible relative Rota-Baxter operators under a suitable condition. Finally, we show the equivalence between r -matrix-Nijenhuis structures and Rota-Baxter-Nijenhuis structures on Lie-Yamaguti algebras. [ABSTRACT FROM AUTHOR]

In this paper, we show the relations between the electrical Lie algebras and the Schrödinger algebras. In particular, we show that the electrical Lie algebra of type F 4 is isomorphic to the Schrödinger algebra in (5 + 1) -dimensional spacetime, which consequently shows that the dimension conjecture for e F 4 holds. Then we classify all simple highest weight modules, all simple Whittaker modules at nonzero level and all simple U (H) -free modules for the Schrödinger algebra in (n + 1) dimensional spacetime. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a finite Lie conformal superalgebra called the Heisenberg-Virasoro Lie conformal superalgebra s by using a class of Heisenberg-Virasoro Lie conformal modules. The super Heisenberg-Virasoro algebra of Ramond type S is defined by the formal distribution Lie superalgebra of s. Then we construct a class of simple S -modules, which are induced from simple modules of some finite dimensional solvable Lie superalgebras. These modules are isomorphic to simple restricted S -modules, and include the highest weight modules, Whittaker modules and high order Whittaker modules. As a byproduct, we present a subalgebra of S , which is isomorphic to the super Heisenberg-Virasoro algebra of Neveu-Schwarz type. [ABSTRACT FROM AUTHOR]

In this paper we prove that any local automorphism on the solvable Leibniz algebras with null-filiform and naturally graded non-Lie filiform nilradicals, whose dimension of complementary space is maximal is an automorphism. Furthermore, the same problem concerning 2-local automorphisms of such algebras is investigated and we obtain the analogously results for 2-local automorphisms. [ABSTRACT FROM AUTHOR]

In this paper, we are interested to study non-abelian extensions of n -Lie algebra. More precisely, we focus to construct the isomorphism theorem of non-abelian extensions. At last, we demonstrate that isomorphism classes of non-abelian extensions of n -Lie algebra are described using equivalence classes of Maurer-Cartan elements in a DGLA. [ABSTRACT FROM AUTHOR]

In this paper, we construct and study a new class of modules over the Block algebra B (q) , where q is a nonzero complex number. We determine the irreducibilities of these modules and the isomorphisms among them. We also show that these modules exhaust all U (h) -free B (q) -modules of rank-1. [ABSTRACT FROM AUTHOR]

This paper is devoted to the complete algebraic and geometric classification of complex 4-dimensional nilpotent left-symmetric algebras. The corresponding geometric variety has dimension 15 and decomposes into 3 irreducible components determined by the Zariski closures of two one-parameter families of algebras and a two-parameter family of algebras (see Theorem B). In particular, there are no rigid 4-dimensional complex nilpotent left symmetric algebras. [ABSTRACT FROM AUTHOR]

In the present paper, we introduce a class of infinite Lie conformal superalgebras S (p) , which are closely related to Lie conformal algebras of extended Block type defined in [6]. Then all finite non-trivial irreducible conformal modules over S (p) for p ∈ C ⁎ are completely classified. As an application, we also present the classifications of finite non-trivial irreducible conformal modules over finite quotient algebras s (n) for n ≥ 1 and sh which is isomorphic to a subalgebra of Lie conformal algebra of N = 2 superconformal algebra. Moreover, as a generalized version of S (p) , the infinite Lie conformal superalgebras GS (p) are constructed, which have a subalgebra isomorphic to the finite Lie conformal algebra of N = 2 superconformal algebra. [ABSTRACT FROM AUTHOR]