Free Akivis Algebras, Primitive Elements, and Hyperalgebras

Free Akivis algebras and primitive elements in their universal enveloping algebras are investigated. It is proved that subalgebras of free Akivis algebras are free and that finitely generated subalgebras are finitely residual. Decidability of the word problem for the variety of Akivis algebras is also proved. The conjecture of K. H. Hofmann and K. Strambach (Problem 6.15 in [Topological and analytic loops, in “Quasigroups and Loops Theory and Applications,” Series in Pure Mathematics (O. Chein, H. O. Pflugfelder, and J. D. H. Smith, Eds.), Vol. 8, pp. 205–262, Heldermann Verlag, Berlin, 1990]) on the structure of primitive elements is proved to be not valid, and a full system of primitive elements in free nonassociative algebra is constructed. Finally, it is proved that every algebra B can be considered as a hyperalgebra, that is, a system with a series of multilinear operations that plays a role of a tangent algebra for a local analytic loop, where the hyperalgebra operations on B are interpreted by certain primitive elements.