nLab Zinbiel algebra

Zinbiel is a mathematical person (in the sense like Nicolas Bourbaki whose mirror person is Leibniz.

A Zinbiel algebra or dual Leibniz algebra (in $Vec$) is a nonassociative algebra with a product satisfying

$$(x y) z = x (z y + y z),$$

Equivalently, it is an algebra over the operad which is Koszul dual to the quadratic operad of Leibniz algebras.

It is also natural to consider dg-algebras over a dg-version of the operad of dual Leibniz algebras.

Dually, a Zinbiel coalgebra is a vector space $C$ equipped with a linear map $\delta:C\to C\otimes C$ satisfying: $\delta\otimes id)\circ\delta = (id\otimes \delta) \circ \delta + (id\otimes \tau\delta)\circ\delta$ where $\tau$ is the flip of tensor factors.

• wikipedia Zinbiel algebra

• Guillaume W. Zinbiel, Encyclopedia of types of algebras 2010, in Guo, Li; Bai, Chengming; Loday, Jean-Louis, Operads and universal algebra, Nankai Series in Pure, Applied Mathematics and Theoretical Physics 9, pp. 217–298, arxiv/1101.0267

• Emily Burgunder, A symmetric version of Kontsevich graph complex and Leibniz homology,
  Journal of Lie Theory 20 (2010), No. 1, 127–165 arxiv/0804.2052

• J-L. Loday, Algebraic K-theory and the conjectural Leibniz K-theory, K-Theory 09/2003; 30(2):105-127, pdf doi

Manon Didry, Construction of groups associated to Lie- and to Leibniz-algebras, Journal of Lie Theory 17 (2007), No. 2, 399–426

We describe a method for associating to a Lie algebra $\mathfrak{g}$ over a ring $K$ a sequence of groups $(G_{n}(\mathfrak{g}))_{n\in\N}$, which are {\it polynomial groups} in the sense that will be explained in Definition 5.1. Using a description of these groups by generators and relations, we prove the existence of an action of the symmetric group $\Sigma_{n}$ by automorphisms. The subgroup of fixed points under this action, denoted by $J_{n}(\mathfrak{g})$, is still a polynomial group and we can form the projective limit $J_{\infty}(\mathfrak{g})$ of the sequence $(J_{n}(\mathfrak{g}))_{n\in\N}$. The formal group $J_{\infty}(\g)$ associated in this way to the Lie algebra $\mathfrak{g}$ may be seen as a generalisation of the formal group associated to a Lie algebra over a field of characteristic zero by the Campbell-Haussdorf formula.