The dual Friedrichs' theorem
V is a vector space over a field of characteristic zero. Friedrichs' theorem characterizes the elements of the Lie subalgebra generated by V in the free associative algebra TV (with its Lie algebra structure \(TV^-\) under the bracket product). The author characterizes the elements of the cofree coassociative coalgebra TcV which are in the Lie coideal J cogenerated by the kernel of \(TcV^-\to V\), where \(TcV^-\) is the Lie coalgebra structure on TcV under the bracket coproduct. The cofree Lie coalgebra LcV and the cofree covered Lie algebra \(TcV^-/J\) are filtered, and the arguments use the associated graded Lie coalgebras. Along the way, it is shown that both of these associated graded Lie coalgebras are isomorphic to the locally nilpotent Lie coalgebra on V, and if V is finite-dimensional, both are isomorphic to the graded dual Lie coalgebra of the free Lie algebra on \(V^*\). The author also identifies the elements of LcV within NcV, the cofree noncoassociative, noncounital coalgebra on V as certain formal infinite sums of Lie elements.
scientific article; zbMATH DE number 4150460