Search Results (6)

We show that a $P{L}_{\infty }$-algebra V can be described by a nilpotent coderivation of degree $-1$ on coalgebra $P^{*}V$. Based on this result, we can generalise the result of Lada to show that every $A_{\infty }$-algebra carries a $P{L}_{\infty }$-algebra structure and every $P{L}_{\infty }$-algebra carries an $L_{\infty }$-algebra structure. In particular, we obtain a pre-Lie n-algebra structure on an arbitrary partially associative n-algebra and deduce that pre-Lie n-algebras are n-Lie admissible. Full article

This study introduces the notions of BiHom-Poisson conformal algebra, BiHom-pre-Poisson conformal algebra, and their related structures. We show that many new BiHom-Poisson conformal algebras can be constructed from a BiHom-Poisson conformal algebra. In particular, the direct product of two BiHom-Poisson conformal algebras is also a BiHom-Poisson conformal algebra. We further describe the conformal bimodule and representation theory of the BiHom-Poisson conformal algebra. In addition, we define BiHom-pre-Poisson conformal algebra as the combination of BiHom-pre-Lie conformal algebra and BiHom-dendriform conformal algebra under some compatibility conditions. We further demonstrate a way to construct BiHom-Poisson conformal algebra from BiHom-pre-Poisson conformal algebra and provide the representation theory for BiHom-pre-Poisson conformal algebra. Finally, a detailed description of $\mathcal{O}$-operators and Rota–Baxter operators on BiHom-Poisson conformal algebra is provided. Full article

The goal of the present paper is to provide a cohomology theory and crossed modules of modified Rota–Baxter pre-Lie algebras. We introduce the notion of a modified Rota–Baxter pre-Lie algebra and its bimodule. We define a cohomology of modified Rota–Baxter pre-Lie algebras with coefficients in a suitable bimodule. Furthermore, we study the infinitesimal deformations and abelian extensions of modified Rota–Baxter pre-Lie algebras and relate them with the second cohomology groups. Finally, we investigate skeletal and strict modified Rota–Baxter pre-Lie 2-algebras. We show that skeletal modified Rota–Baxter pre-Lie 2-algebras can be classified into the third cohomology group, and strict modified Rota–Baxter pre-Lie 2-algebras are equivalent to the crossed modules of modified Rota–Baxter pre-Lie algebras. Full article

In this paper, we first introduce the notion of a 3-Hom–Lie bialgebra and give an equivalent description of the 3-Hom–Lie bialgebras, the matched pairs and the Manin triples of 3-Hom–Lie algebras. In addition, we define $\mathcal{O}$-operators of 3-Hom–Lie algebras and construct solutions of the 3-Hom–Lie Yang–Baxter equation in terms of $\mathcal{O}$-operators and 3-Hom–pre-Lie algebras. Finally, we show that a 3-Hom–Lie algebra has a phase space if and only if it is sub-adjacent to a 3-Hom–pre-Lie algebra. Full article

The Lie algebraic scheme for constructing Hamiltonian operators is differential-algebraically recast and an effective approach is devised for classifying the underlying algebraic structures of integrable Hamiltonian systems. Lie–Poisson analysis on the adjoint space to toroidal loop Lie algebras is employed to construct new reduced pre-Lie algebraic structures in which the corresponding Hamiltonian operators exist and generate integrable dynamical systems. It is also shown that the Balinsky–Novikov type algebraic structures, obtained as a Hamiltonicity condition, are derivations on the Lie algebras naturally associated with differential toroidal loop algebras. We study nonassociative and noncommutive algebras and the related Lie-algebraic symmetry structures on the multidimensional torus, generating via the Adler–Kostant–Symes scheme multi-component and multi-dimensional Hamiltonian operators. In the case of multidimensional torus, we have constructed a new weak Balinsky–Novikov type algebra, which is instrumental for describing integrable multidimensional and multicomponent heavenly type equations. We have also studied the current algebra symmetry structures, related with a new weakly deformed Balinsky–Novikov type algebra on the axis, which is instrumental for describing integrable multicomponent dynamical systems on functional manifolds. Moreover, using the non-associative and associative left-symmetric pre-Lie algebra theory of Zelmanov, we also explicate Balinsky–Novikov algebras, including their fermionic version and related multiplicative and Lie structures. Full article