# Anti-commutative algebra

From Encyclopedia of Mathematics

A linear algebra over a field in which the identity

\begin{equation}x^2=0\label{*}\end{equation}

is valid. If the characteristic of the field differs from 2, the identity \eqref{*} 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=43603

This article was adapted from an original article by A.T. Gainov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article