On Enveloping Lie Algebras of Hyporeductive Triple Algebras

Abstract

The notion of a hypoderivation of binary-ternary algebras is introduced. A hypoderivation is a generalization both of a derivation and a pseudoderivation of such algebras. From the external direct sum of a hyporeductive triple algebra (h.t.a.) with the vector space of pairs constituted by hypoderivations and their companions, a Lie algebra with a hyporeductive decomposition (and accordingly a hyporeductive pair) enveloping the given h.t.a. is constructed. A nontrivial 3-dimensional Lie algebra with hyporeductive decomposition is presented. Examples of h.t.a. are also given.

Key Words:

• Akivis algebra
• Bol algebra
• Hyporeductive triple algebra
• Lie triple algebra (generalized Lie triple system)
• Lie triple system
• Smooth loop

2000 Mathematics Subject Classification:

• Primary 17D99
• Secondary 20N05

ACKNOWLEDGMENT

I wish to thank the referee for suggesting improvements to many parts of this paper and for carefully reading the paper.

Notes

Communicated by E. Zelmanov.