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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 VecVec) is a nonassociative algebra with a product satisfying

(xy)z=x(zy+yz), (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 CC equipped with a linear map δ:CCC\delta:C\to C\otimes C satisfying: δid)δ=(idδ)δ+(idτδ)δ\delta\otimes id)\circ\delta = (id\otimes \delta) \circ \delta + (id\otimes \tau\delta)\circ\delta where τ\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 𝔤\mathfrak{g} over a ring KK a sequence of groups (G n(𝔤)) nN(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 Σ n\Sigma_{n} by automorphisms. The subgroup of fixed points under this action, denoted by J n(𝔤)J_{n}(\mathfrak{g}), is still a polynomial group and we can form the projective limit J (𝔤)J_{\infty}(\mathfrak{g}) of the sequence (J n(𝔤)) nN(J_{n}(\mathfrak{g}))_{n\in\N}. The formal group J (g)J_{\infty}(\g) associated in this way to the Lie algebra 𝔤\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.

[[!redirect Zinbiel coalgebra]]

Last revised on December 13, 2013 at 08:09:43. See the history of this page for a list of all contributions to it.