Commutant-associative algebra (original) (raw)

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In abstract algebra, a commutant-associative algebra is a nonassociative algebra over a field whose multiplication satisfies the following axiom:

( [ A 1 , A 2 ] , [ A 3 , A 4 ] , [ A 5 , A 6 ] ) = 0 {\displaystyle ([A_{1},A_{2}],[A_{3},A_{4}],[A_{5},A_{6}])=0} {\displaystyle ([A_{1},A_{2}],[A_{3},A_{4}],[A_{5},A_{6}])=0},

where [A, _B_] = AB − BA is the commutator of A and B and (A, B, C) = (AB)C – A(BC) is the associator of A, B and C.

In other words, an algebra M is commutant-associative if the commutant, i.e. the subalgebra of M generated by all commutators [A, _B_], is an associative algebra.