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Difference between revisions of "Anti-commutative algebra"

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A linear algebra over a field in which the identity
 
A linear algebra over a field in which the identity
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012580/a0125801.png" /></td> <td valign="top" style="width:5%;text-align:right;">(*)</td></tr></table>
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\begin{equation}x^2=0\label{*}\end{equation}
  
is valid. If the characteristic of the field differs from 2, the identity (*) is equivalent with the identity <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012580/a0125802.png" />. 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]]).
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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|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>

Latest revision as of 16:53, 30 December 2018

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://encyclopediaofmath.org/index.php?title=Anti-commutative_algebra&oldid=12204
This article was adapted from an original article by A.T. Gainov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article