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ABSTRACT
The free nonassociative algebra has two subspaces which are closed under both the commutator and the associator: the Akivis elements and the primitive elements. Every Akivis element is primitive, but there are primitive elements which are not Akivis. Using a theorem of Shestakov, we give a recursive formula for the dimension of the Akivis elements. Using a theorem of Shestakov and Umirbaev, we prove a closed formula for the dimension of the primitive elements. These results generalize the Witt dimension formula for the Lie elements in the free associative algebra.
Mathematics Subject Classification:
ACKNOWLEDGMENTS
Murray Bremner thanks NSERC for financial support, the University of Saskatchewan for a sabbatical leave from January to June 2004, and the Departments of Mathematics at Iowa State University and the University of São Paulo for their hospitality.
This article was written while Luiz Peresi held grants from CNPq of Brazil and FAPESP. Part of this research was done when this author was visiting Iowa State University on a grant from FAPESP.
We thank Emeric Deutsch for references Deutsch and Shapiro (Citation2001a); Deutsch and Shapiro (Citation2001b), and for helpful emails.
Notes
Communicated by I. Shestakov.
