Abstract
We review the developments in the Lie theory for non-associative products from 2000 to date and describe the current understanding of the subject in view of the recent works, many of which use non-associative Hopf algebras as the main tool.
Highlights
Lie groups, Lie algebras and Hopf algebrasNon-associative Lie theory, that is, Lie theory for non-associative products, appeared as a a subject of its own in the works of Malcev who constructed the tangent structures corresponding to Moufang loops
Given all that has been said about the flat Sabinin algebras, it is not hard to show that Sabinin algebras form a category which is equivalent to the category of all formal loops
It turned out that a similar construction can be carried out for Bol algebras [45] and, more generally, all Sabinin algebras [46]. Another motivation for the development of the machinery of non-associative Hopf algebras was the question of whether the commutator and the associator are the only primitive operations in a non-associative bialgebra. It appeared as a conjecture in [20]; if it were true, it would imply an important role for the Akivis algebras in non-associative Lie theory
Summary
Non-associative Lie theory, that is, Lie theory for non-associative products, appeared as a a subject of its own in the works of Malcev who constructed the tangent structures corresponding to Moufang loops. For finite-dimensional Lie algebras, the Baker–Campbell–Hausdorff series always has a non-zero radius of convergence and, defines a local Lie group. According to the Poincaré–Birkhoff–Witt theorem, as a coalgebra U (g) is isomorphic to the symmetric algebra k[g] This latter can be thought of as the universal enveloping algebra of the abelian Lie algebra which coincides with g as a vector space and whose Lie bracket is identically zero. It is not hard to prove that it gives an equivalence of the category of cocommutative primitively generated Hopf algebras to that of Lie algebras (of not necessarily finite dimension). To recover a Lie group from a Hopf algebra, one uses the fact that any cocommutative Hopf algebra generated by its primitive elements is a universal enveloping algebra of some Lie algebra. This map is, a finite-dimensional local Lie group and these are always locally equivalent to -connected Lie groups
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