A linear algebra over a field in which the identity
is valid. If the characteristic of the field differs from 2, the identity (*) is equivalent with the identity $xy=-yx$. All subalgebras of a free anti-commutative algebra are free. The most important varieties of anti-commutative algebras are Lie algebras, Mal'tsev algebras and binary Lie algebras (cf. Lie algebra; Binary Lie algebra; Mal'tsev algebra).
|||A.I. Shirshov, "Subalgebras of free commutative and free anti-commutative algebras" Mat. Sb. , 34 (76) : 1 (1954) pp. 81–88 (In Russian)|
Anti-commutative algebra. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Anti-commutative_algebra&oldid=31731