Let A be a graded-commutative, connected k-algebra generated in degree 1. The homotopy Lie algebra gA is defined to be the Lie algebra of primitives of the Yoneda algebra, ExtA(k, k). Under certain homological assumptions on A and its quadratic closure, we express gA as a semi-direct product of the well-understood holonomy Lie algebra hA with a certain hA-module. This allows us to compute the h...

We study the Hopf algebra H of Fliess operators coming from Control Theory in the one-dimensional case. We prove that it admits a graded, finte-dimensional, connected gradation. Dually, the vector space R〈x0, x1〉 is both a pre-Lie algebra for the pre-Lie product dual of the coproduct of H, and an associative, commutative algebra for the shuffle product. These two structures admit a compatibilit...

In this paper, first we introduce the notions of relative Rota-Baxter operators nonzero weight on $3$-Lie algebras and $3$-post-Lie algebras. A 3-post-Lie algebra consists a 3-Lie structure ternary operation such that some compatibility conditions are satisfied. We show operator induces naturally. Conversely, gives rise to new algebra, which is called subadjacent an action original algebra. The...

We study the existence problem for Novikov algebra structures on finite-dimensional Lie algebras. We show that a Lie algebra admitting a Novikov algebra is necessarily solvable. Conversely we present a 2-step solvable Lie algebra without any Novikov structure. We use extensions and classical r-matrices to construct Novikov structures on certain classes of solvable Lie algebras.

‎‎we exhibit an explicit construction for the second cohomology group‎ ‎$h^2(l‎, ‎a)$ for a lie ring $l$ and a trivial $l$-module $a$‎. ‎we show how the elements of $h^2(l‎, ‎a)$ correspond one-to-one to the‎ ‎equivalence classes of central extensions of $l$ by $a$‎, ‎where $a$‎ ‎now is considered as an abelian lie ring‎. ‎for a finite lie‎ ‎ring $l$ we also show that $h^2(l‎, ‎c^*) cong m(l)$‎...

Abstract. The underlying field is of characteristic p > 2. In this paper, we first use the method of computing the homogeneous derivations to determine the first cohomology of the so-called odd contact Lie algebra with coefficients in the even part of the generalized Witt Lie superalgebra. In particular, we give a generating set for the Lie algebra under consideration. Finally, as an applicatio...

We consider three Lie algebras: Der C((t)), the Lie algebra of all derivations on the algebra C((t)) of formal Laurent series; the Lie algebra of all differential operators on C((t)); and the Lie algebra of all differential operators on C((t)) ⊗ Cn. We prove that each of these Lie algebras has an essentially unique nontrivial central extension. The Lie algebra of all derivations on the Laurent ...