# Anti-commutative algebra

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A linear algebra over a field in which the identity

$$x^2=0\tag{*}$$

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).

#### References

 [1] A.I. Shirshov, "Subalgebras of free commutative and free anti-commutative algebras" Mat. Sb. , 34 (76) : 1 (1954) pp. 81–88 (In Russian)
How to Cite This Entry:
Anti-commutative algebra. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Anti-commutative_algebra&oldid=31731
This article was adapted from an original article by A.T. Gainov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article