A non-associative algebra is power-associative if, for any cyclic subalgebra of , where is the associator.
If we inductively define the powers of an element by
(when is unital with ) ,
then power-associativity of means that for any non-negative integers and , since the associator is trilinear (linear in each of the three coordinates). This implies that . In addition, .
A theorem, due to A. Albert, states that any finite power-associative division algebra over the integers of characteristic not equal to 2, 3, or 5 is a field. This is a generalization of the Wedderburn’s Theorem on finite division rings.
|Date of creation||2013-03-22 14:43:27|
|Last modified on||2013-03-22 14:43:27|
|Last modified by||CWoo (3771)|