The dual Friedrichs' theorem
type: Item
instance of: scholarly article
title: The dual Friedrichs' theorem
zbMATH Open document ID: 0702.17015
DOI: 10.1016/0021-8693(90)90140-J
author: Gary Griffing
published in: Journal of Algebra
publication date: 1990-01-01T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 17B99
Mathematics Subject Classification ID: 16W30
Mathematics Subject Classification ID: 16S10
Mathematics Subject Classification ID: 17B01
Mathematics Subject Classification ID: 16W10
zbMATH DE Number: 4150460
zbMATH Keywords: free associative algebra
zbMATH Keywords: cofree coassociative coalgebra
zbMATH Keywords: cofree Lie coalgebra
zbMATH Keywords: cofree covered Lie algebra
zbMATH Keywords: graded dual Lie coalgebra
zbMATH Keywords: free Lie algebra
reviewed by: Earl J. Taft
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 4150460