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In this paper, we mainly present some new solutions of the Hom-Yang-Baxter equation from Hom-algebras, Hom-coalgebras and Hom-Lie algebras, respectively. Also, we prove that these solutions are all self-inverse and give some examples. Finally, we introduce the notion of Hom-Yang-Baxter systems and obtain two kinds of Hom-Yang-Baxter systems., Comment: 20

In a chimera state, domains composed of synchronized oscillators coexist with ones composed of desynchronized oscillators. It is a common view that, in a chimera state, oscillators in coherent domains always share the same mean phase velocity. However, recent studies have suggested that oscillators in different coherent domains may have different mean phase velocities. In this work, we study a ring of nonlocally coupled Brusselators. We find a two-frequency chimera state with mixed phase regularities in which Brusselators in adjacent coherent domains oscillate at different velocities. Moreover, Brusselators in coherent domains with higher mean phase velocity are nearly in phase. In contrast, Brusselators in coherent domains with lower mean phase velocity are randomly partitioned into two groups in antiphase. We find that the local mean fields in these two types of coherent domains perform different dynamics. Based on local mean fields, we provide an explanation for the formation of this type of chimera state. Furthermore, the stability diagrams of the two-frequency chimera state with mixed phase regularities are investigated in different parameter planes

In this paper, we first discuss cohomology and a one-parameter formal deformation theory of Lie–Yamaguti algebras. Next, we study finite group actions on Lie–Yamaguti algebras and introduce equivariant cohomology for Lie–Yamaguti algebras equipped with group actions. Finally, we study an equivariant one-parameter formal deformation theory and show that our equivariant cohomology is the suitable deformation cohomology.

The goal of this paper is to examine the structure of split involutive regular BiHom-Lie superalgebras, which can be viewed as the natural generalization of split involutive regular Hom-Lie algebras and split regular BiHom-Lie superalgebras. By developing techniques of connections of roots for this kind of algebras, we show that such a split involutive regular BiHom-Lie superalgebra L {\mathfrak{L}} is of the form L = U + ∑ α I α {\mathfrak{L}}=U+{\sum }_{\alpha }{I}_{\alpha } with U a subspace of a maximal abelian subalgebra H and any I α , a well-described ideal of L {\mathfrak{L}} , satisfying [I α , I β ] = 0 if [α] ≠ [β]. In the case of L {\mathfrak{L}} being of maximal length, the simplicity of L {\mathfrak{L}} is also characterized in terms of connections of roots.

In this paper, we study 3-Lie algebras with derivations. We call the pair consisting of a 3-Lie algebra and a distinguished derivation by the 3-LieDer pair. We define a cohomology theory for 3-LieDer pair with coefficients in a representation. We study central extensions of a 3-LieDer pair and show that central extensions are classified by the second cohomology of the 3-LieDer pair with coefficients in the trivial representation. We generalize Gerstenhaber's formal deformation theory to 3-LieDer pairs in which we deform both the 3-Lie bracket and the distinguished derivation., Comment: 18 pages. arXiv admin note: text overlap with arXiv:2003.07392 by other authors

A cohomology theory of weighted Rota-Baxter $3$-Lie algebras is introduced. Formal deformations, abelian extensions, skeletal weighted Rota-Baxter $3$-Lie 2-algebras and crossed modules of weighted Rota-Baxter 3-Lie algebras are interpreted by using lower degree cohomology groups., Comment: Title changed to "Deformations and cohomology theory of Rota-Baxter $3$-Lie algebras of arbitrary weights" J. Geom. Phys. to appear

The aim of this paper is to study the structures of split regular Hom-Lie Rinehart algebras. Let $(L,A)$ be a split regular Hom-Lie Rinehart algebra. We first show that $L$ is of the form $L=U+\sum_{[\gamma]\in\Gamma/\thicksim}I_{[\gamma]}$ with $U$ a vector space complement in $H$ and $I_{[\gamma]}$ are well described ideals of $L $ satisfying $I_{[\gamma]},I_{[\delta]}=0$ if $I_{[\gamma]}\neq I_{[\delta]}$. Also, we discuss the weight spaces and decompositions of $A$ and present the relation between the decompositions of $L$ and $A$. Finally, we consider the structures of tight split regular Hom-Lie Rinehart algebras.

Let (H,?) be a Hom-Hopf algebra and (A,?) be a Hom-algebra. In this paper we will construct the Hom-crossed product (A#?H???), and prove that the extension A ? A#?H is actually a Hom-type cleft extension and vice versa. Then we will give the necessary and sufficient conditions to make (A#?H???) into a Hom-Hopf algebra. Finally we will study the lazy 2-cocycle on (H,?).

The paper introduces the class of split regular BiHom-Poisson superalgebras, which is a natural generalization of split regular Hom-Poisson algebras and split regular BiHom-Lie superalgebras. By developing techniques of connections of roots for this kind of algebras, we show that such a split regular BiHom-Poisson superalgebras $A$ is of the form $A=U+\sum_{\a}I_\a$ with $U$ a subspace of a maximal abelian subalgebra $H$ and any $I_{\a}$, a well described ideal of $A$, satisfying $[I_\a, I_\b]+I_\a I_\b = 0$ if $[\a]\neq [\b]$. Under certain conditions, in the case of $A$ being of maximal length, the simplicity of the algebra is characterized., Comment: 16 pages. arXiv admin note: text overlap with arXiv:1508.02124, arXiv:1706.07084 by other authors

Let $(H,\alpha)$ be a monoidal Hom-Hopf algebra and $^{H}_{H}\mathcal{HYD}$ the Hom-Yetter-Drinfeld category over $(H,\alpha)$. Then in this paper, we first introduce the definition of braided Hom-Lie algebras and show that each monoidal Hom-algebra in $^{H}_{H}\mathcal{HYD}$ gives rise to a braided Hom-Lie algebra. Second, we prove that if $(A,\beta)$ is a sum of two $H$-commutative monoidal Hom-subalgebras, then the commutator Hom-ideal $[A,A]$ of $A$ is nilpotent. Also, we study the central invariant of braided Hom-Lie algebras as a generalization of generalized Lie algebras. Finally, we obtain a construction of the enveloping algebras of braided Hom-Lie algebras and show that the enveloping algebras are $H$-cocommutative Hom-Hopf algerbas., Comment: 31pages

Chimera states refer to the dynamical states in which the inherent symmetry of the system is broken. The system composed of two interacting identical subpopulations of phase oscillators provides a platform to study chimera states. In this system, different types of chimera states have been identified and the transitions between them have been investigated. However, the parameter space is not fully explored in this system. In this work, we study a system comprised of two interacting subpopulations of nonidentical phase oscillators. Through numerical simulations and theoretical analyses, we find three symmetry-reserving states, including incoherent state, in-phase synchronous state, and antiphase synchronous state, and three types of symmetry-breaking states, including in-phase chimera states, antiphase chimera states, and weak chimera states. The stability diagrams of these dynamical states are explored on different parameter planes and transition scenarios amongst these states are investigated. We find that the weak chimera states act as the bridge between in-phase and antiphase chimera states. We also observe the existence of a period-two chimera state, chaotic chimera state, and drifting chimera states.

In this paper, we first introduce the notion of BiHom–Lie superalgebras, which is a generalization of both BiHom–Lie algebras and Hom–Lie superalgebras. Also, we explore some general classes of BiHom–Lie admissible superalgebras and describe all these classes via G-BiHom-associative superalgebras, where G is a subgroup of the symmetric group $$S_{3}$$ . Finally, we obtain a method to construct the solutions of the BiHom–Yang–Baxter equation from BiHom–Lie superalgebras.

We find a sufficient condition for the category of entwined Hom-modules to be monoidal. Moreover, we introduce morphisms between the underlying monoidal Hom-algebras and monoidal Homcoalgebras, which give rise to functors between the category of entwined Hom-modules, and we study tensor identities for monodial categories of entwined Hom-modules. Finally, we give necessary and sufficient conditions for the general induction functor from ~H (Mk)(ψ)CA to ~H (Mk)(ψ')C'A' to be separable.

Entwined Hom-modules were introduced by Karacuha in [13], which can be viewed as a generalization of Doi-Hom Hopf modules and entwined modules. In this paper, the sufficient and necessary conditions for the forgetful functor F : ?H(Mk)(?)CA ? ?H(Mk)A and its adjoint G : ?H(Mk)A ? ?H (Mk)(?)CA form a Frobenius pair are obtained, one is that A?C and the C*?A are isomorphic as (A;C*op#A)-bimodules, where (A,C,?) is a Hom-entwining structure. Then we can describe the isomorphism by using a generalized type of integral. As an application, a Maschke type theorem for entwined Hom-modules is given.

Spiral wave chimeras with spatially extended unsynchronized cores have been reported in nonlocally coupled phase oscillators and chaotic oscillators. Here, we investigate the two-dimensional nonlocally coupled FitzHugh–Nagumo oscillators with open boundary conditions. Rich dynamics of spiral wave chimeras is discovered numerically. Besides outwardly propagating and inwardly propagating single-spiral wave chimeras, we find the double-spiral wave chimera with two spiral chimeras rotating with each other and the breakup of spiral wave chimeras. We also find a transition between a normal spiral wave with a phase singularity at its center and spiral wave chimera.

In this work, we study the Sakaguchi-Kuramoto model with natural frequency following a bimodal distribution. By using Ott-Antonsen ansatz, we reduce the globally coupled phase oscillators to low dimensional coupled ordinary differential equations. For symmetrical bimodal frequency distribution, we analyze the stabilities of the incoherent state and different partial synchronous states. Different types of bifurcations are identified and the effect of the phase lag on the dynamics is investigated. For asymmetrical bimodal frequency distribution, we observe the revival of the incoherent state, and then the conditions for the revival are specified.

We introduce the notion of 3-Hom-Lie-Rinehart algebra and systematically describe a cohomology complex by considering coefficient modules. Furthermore, we consider extensions of a 3-Hom-Lie-Rinehart algebra and characterize the first cohomology space in terms of the group of automorphisms of an $A$-split abelian extension and the equivalence classes of $A$-split abelian extensions. Finally, we study formal deformations of 3-Hom-Lie-Rinehart algebras., 16 pages. arXiv admin note: substantial text overlap with arXiv:1808.01909 by other authors

The main purpose of the present paper is to develop the theory of center constructions on Hom–Hopf algebras. Let H be a Hom–Hopf algebra, we first introduce the notions of nth Yetter–Drinfeld modules and mth Drinfeld codouble for H. Also we prove that the category $${\mathcal {YD}}_H^H(n)$$ of nth Yetter–Drinfeld modules of H is a braided autonomous category. Finally, we show that $${\mathcal {YD}}_H^H(n)$$ and $$Corep^{i,j}(CD_m(H))$$ (i.e., the corepresentation category of the Drinfeld codouble of H) are braided isomorphic as the full subcategories of $$Corep^{i,j}(H)$$ .

In this paper, we construct a new example of Hopf group coalgebras by considering Radford's biproduct Hopf algebra in the Turaev category. Furthermore, we find some sufficient and necessary conditions for such Radford's biproduct Hopf algebra to admit quasitriangular structures in the sense of Turaev group coalgebras.

The goal of this paper is to study the structure of split regular BiHom-Leiniz superalgebras, which is a natural generalization of split regular Hom-Leiniz algebras and split regular BiHom-Lie superalgebras. By developing techniques of connections of roots for this kind of algebras, we show that such a split regular BiHom-Leiniz superalgebras $\mathfrak{L}$ is of the form $\mathfrak{L}=U+\sum_{\a}I_\a$ with $U$ a subspace of a maximal abelian subalgebra $H$ and any $I_{\a}$, a well described ideal of $\mathfrak{L}$, satisfying $[I_\a, I_\b]= 0$ if $[\a]\neq [\b]$. In the case of $\mathfrak{L}$ being of maximal length, the simplicity of $\mathfrak{L}$ is also characterized in terms of connections of roots., Comment: 16pages. arXiv admin note: substantial text overlap with arXiv:1902.06260; text overlap with arXiv:1504.04236, arXiv:1508.02124 by other authors

In this paper, we give the definitions of crossed quasi-Hopf π-coalgebra H and a crossed left π-H-modules, and show that the category of crossed left π-H-modules is a monoidal category. Finally, we show that a family R = {R α, β ∈ H α ⊗ H β } of elements is a quasitriangular structure of a crossed quasi-Hopf π-coalgebra H if and only if the category of crossed left π-H-modules over H is a braided monoidal category with braiding defined by R.

We obtain necessary and sufficient conditions for the functor [Formula: see text] on the category of partial entwined modules that forgets the A-action to be separable. As an application, we prove a Maschke-type theorem for the category of partial entwined modules.

In this paper, we first introduce the notion of Hom-left-symmetric conformal bialgebras and show some nontrivial examples. Also, we present construction methods of matched pairs of Hom-Lie conformal algebras and Hom-left-symmetric conformal algebras. Finally, we prove that a finite Hom-left-symmetric conformal bialgebra is free as a $\mathbb{C}[\partial]$-module is equivalent to a Hom-parak\"{a}hler Lie conformal algebra. In particular, we investigate the coboundary Hom-left-symmetric conformal bialgebras., Comment: 20

We first introduce the notion of Doi Hom-Hopf modules and find the sufficient condition for the category of Doi Hom-Hopf modules to be monoidal. Also we obtain the condition for the monoidal Hom-algebra and monoidal Hom-coalgebra to be monoidal Hom-bialgebras. Second, we give the maps between the underlying monoidal Hom-Hopf algebras, Hom-comodule algebras and Hom-module coalgebras give rise to functors between the category of Doi Hom-Hopf modules and study tensor identities for monodial categories of Doi Hom-Hopf modules. Furthermore, we construct a braiding on the category of Doi Hom-Hopf modules. Finally, as an application of our theory, we consider the braiding on the category of Hom-modules, the category of Hom-comodules and the category of Hom-Yetter-Drinfeld modules respectively.

Let H be a weak Hopf π-coalgebra and let A be a weak right π-H-comodule algebra with a total integral ϕ. This paper is mainly devoted to a Maschke type theorem for (H,A)-weak relative Hopf π-modules. As an application, we obtain a characterization for an (H,A)-weak Hopf π-module to be projective as an A-module.

Let (H, α) be a monoidal Hom-Hopf algebra and (A, β) a right (H, α)-Homcomodule algebra. We first introduce the notion of a relative Hom-Hopf module and prove that the functor F from the category of relative Hom-Hopf modules to the category of right (A, β)-Hom-modules has a right adjoint. Furthermore, we prove a Maschke type theorem for the category of relative Hom-Hopf modules. In fact, we give necessary and sufficient conditions for the functor that forgets the (H, α)-coaction to be separable. This leads to a generalized notion of integrals.

We first introduce the notion of a right generalized partial smash product and explore some properties of such partial smash product, and consider some examples. Furthermore, we introduce the notion of a generalized partial twisted smash product and discuss a necessary condition under which such partial smash product forms a Hopf algebra. Based on these notions and properties, we construct a Morita context for partial coactions of a co-Frobenius Hopf algebra.

Let G be a group. In this paper we investigate the notion of a quasitriangular Hopf G-coalgebra. We construct a class of smash product Hopf G-coalgebras and present a framework for constructing a quasitriangular Hopf G-coalgebra. As an application, we study an analog of the Drinfel'd double for a crossed Hopf G-coalgebra of finite type.

BiHom-Novikov agebra is a generalized Hom-Novikov algebra endowed with two commuting multiplicative linear maps. The main purpose of this paper is to show that two classes of BiHom-Novikov algebras can be constructed from BiHom-commutative algebras together with derivations and BiHom-Novikov algebras with Rota-Baxter operators, respectively. We show that quadratic BiHom-Novikov algebras are associative algebras and the sub-adjacent BiHom-Lie algebras of BiHom-Novikov algebras are 2-step nilpotent. Moreover, we develop the 1-parameter formal deformation theory of BiHom-Novikov algebras., Comment: 19. arXiv admin note: substantial text overlap with arXiv:1501.00229, arXiv:1204.6373 by other authors

Let (H, S, α) be a monoidal Hom-Hopf algebra and \documentclass[12pt]{minimal}\begin{document}$^{H}_{H}\mathcal {HYD}$\end{document}HYDHH the Hom-Yetter-Drinfeld category over (H, α). Then in this paper, we first find sufficient and necessary conditions for \documentclass[12pt]{minimal}\begin{document}$^{H}_{H}\mathcal {HYD}$\end{document}HYDHH to be symmetric and pseudosymmetric, respectively. Second, we study the u-condition in \documentclass[12pt]{minimal}\begin{document}$^{H}_{H}\mathcal {HYD}$\end{document}HYDHH and show that the Hom-Yetter-Drinfeld module (H, adjoint, Δ, α) (resp., (H, m, coadjoint, α)) satisfies the u-condition if and only if S2 = id. Finally, we prove that \documentclass[12pt]{minimal}\begin{document}$^{H}_{H}\mathcal {HYD}$\end{document}HYDHH over a triangular (resp., cotriangular) Hom-Hopf algebra contains a rich symmetric subcategory.

The main aim of this paper is to study Hopf group-crossed products and Hopf group-cleft extensions in the setting of Hopf group-coalgebras.

In this paper, we study symmetric pairs and pseudosymmetries in the Hom-Yetter–Drinfeld category [Formula: see text] over a Hom-Hopf algebra [Formula: see text]. We first show that the category [Formula: see text] over a (co)triangular Hom-Hopf algebra [Formula: see text] contains a rich symmetric subcategory. Also we prove that the (co)commutativity and trivial property of [Formula: see text] are determined by some symmetric pairs of objects in [Formula: see text]. Moreover, we find a sufficient and necessary condition for [Formula: see text] to be pseudosymmetric.

Let [Formula: see text] be a monoidal Hom-Hopf algebra, [Formula: see text] a right [Formula: see text]-Hom-comodule algebra and [Formula: see text] a right [Formula: see text]-Hom-module coalgebra. We first investigate the criterion for the existence of a total integral of [Formula: see text] in the setting of monoidal Hom-Hopf algebras. Also, we prove that there exists a total integral [Formula: see text] if and only if any representation of the pair [Formula: see text] is injective in a functorial way, which generalizes Menini and Militaru’s result. Finally, we extend to the category of [Formula: see text]-Doi Hom-Hopf modules a result of Doi on projectivity of every relative [Formula: see text]-Hopf module as an [Formula: see text]-module.