Primitive elements of a connected free bialgebra
Lo\"ic Foissy (LMPA)
TL;DR
This paper proves that the primitive elements of a graded, connected free bialgebra form a free Lie algebra, using filtration techniques and classical theorems in algebra.
Contribution
It establishes that the Lie algebra of primitive elements in such bialgebras is itself free, extending classical results to a broader algebraic context.
Findings
Primitive elements form a free Lie algebra
Use of filtration embeds graded Lie algebra into free cocommutative bialgebra
Application of Cartier-Quillen-Milnor-Moore and Shirshov-Witt theorems
Abstract
We prove that the Lie algebra of primitive elements of a graded and connected bialgebra, free as an associative algebra, over a eld of characteristic zero, is a free Lie algebra. The main tool is a ltration, which allows to embed the associated graded Lie algebra into the Lie algebra of a free and cocommutative bialgebra. The result is then a consequence of Cartier-Quillen-Milnor-Moore's Shirshov-Witt's theorems.
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Taxonomy
TopicsAdvanced Topics in Algebra · Algebraic structures and combinatorial models · Nonlinear Waves and Solitons