The multigraded Nijenhuis-Richardson Algebra, its universal property and application

n-ary Lie and Associative Algebras

With the help of the multigraded Nijenhuis-- Richardson bracket and the multigraded Gerstenhaber bracket from [7] for every $n\ge 2$ we define $n$-ary associative algebras and their modules and also $n$-ary Lie algebras and their modules, and we give the relevant formulas for Hochschild and Chevalley cohomogy.

Universal Lie Formulas for Higher Antibrackets

We prove that the hierarchy of higher antibrackets (aka higher Koszul brackets, aka Koszul braces) of a linear operator $Δ$ on a commutative superalgebra can be defined by some universal formulas involving iterated Nijenhuis-Richardson brackets having as arguments $Δ$ and the multiplication operators. As a byproduct, we can immediately extend higher antibrackets to noncommutative algebras in a way preserving the validity of generalized Jacobi identities.

Notes on Formal Deformations of Hom-associative and Hom-Lie Algebras

The aim of this paper is to extend to Hom-algebra structures the theory of formal deformations of algebras which was introduced by Gerstenhaber for associative algebras and extended to Lie algebras by Nijenhuis-Richardson. We deal with Hom-associative and Hom-Lie algebras. We construct the first groups of a deformation cohomology and give several examples of deformations. We provide families of Hom-Lie algebras deforming Lie algebra $sl_2$ and describe as formal deformations the q-deformed Witt algebra and Jackson $sl_2$.

扭理论、相对 Rota-Baxter 型算子与 Lie 共形代数上的 $L_\infty$-代数

Twisting theory, relative Rota-Baxter type operators and $L_\infty$-algebras on Lie conformal algebras

本文建立 Lie 共形代数的扭理论，利用导出括号构造 $L_\infty$-代数，刻画相对 Rota-Baxter 型算子，并引入 NS-Lie 共形代数及相关上同调与形变。