Algebra and Logic

, Volume 58, Issue 4, pp 322–326 | Cite as

Associators and Commutators in Alternative Algebras

  • E. Kleinfeld
  • I. P. ShestakovEmail author

It is proved that in a unital alternative algebra A of characteristic ≠ 2, the associator (a, b, c) and the Kleinfeld function f(a, b, c, d) never assume the value 1 for any elements a, b, c, dA. Moreover, if A is nonassociative, then no commutator [a, b] can be equal to 1. As a consequence, there do not exist algebraically closed alternative algebras. The restriction on the characteristic is essential, as exemplified by the Cayley–Dickson algebra over a field of characteristic 2.


alternative algebra associator commutator Kleinfeld function 


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Authors and Affiliations

  1. 1.RenoUSA
  2. 2.Sobolev Institute of MathematicsNovosibirskRussia
  3. 3.Universidade de São PauloSão Paulo-SEPBrazil

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