Difference between revisions of "Anti-commutative algebra"
From Encyclopedia of Mathematics
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A linear algebra over a field in which the identity | 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 | + | 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|Lie algebra]]; [[Binary Lie algebra|Binary Lie algebra]]; [[Mal'tsev algebra|Mal'tsev algebra]]). |
====References==== | ====References==== | ||
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> A.I. Shirshov, "Subalgebras of free commutative and free anti-commutative algebras" ''Mat. Sb.'' , '''34 (76)''' : 1 (1954) pp. 81–88 (In Russian)</TD></TR></table> | <table><TR><TD valign="top">[1]</TD> <TD valign="top"> A.I. Shirshov, "Subalgebras of free commutative and free anti-commutative algebras" ''Mat. Sb.'' , '''34 (76)''' : 1 (1954) pp. 81–88 (In Russian)</TD></TR></table> | ||
Revision as of 15:15, 15 April 2014
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://encyclopediaofmath.org/index.php?title=Anti-commutative_algebra&oldid=31731
Anti-commutative algebra. Encyclopedia of Mathematics. URL: http://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