Namespaces
Variants
Actions

Difference between revisions of "Binary Lie algebra"

From Encyclopedia of Mathematics
Jump to: navigation, search
(Importing text file)
 
m (eqref)
 
(2 intermediate revisions by 2 users not shown)
Line 1: Line 1:
''<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016350/b0163502.png" />-algebra''
+
{{TEX|done}}
 +
''$BL$-algebra''
  
A linear algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016350/b0163503.png" /> over a field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016350/b0163504.png" /> any two elements of which generate a Lie subalgebra. The class of all binary Lie algebras over a given field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016350/b0163505.png" /> generates a variety which, if the characteristic of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016350/b0163506.png" /> is different from 2, is given by the system of identities
+
A linear algebra $A$ over a field $F$ any two elements of which generate a Lie subalgebra. The class of all binary Lie algebras over a given field $F$ generates a variety which, if the characteristic of $F$ is different from 2, is given by the system of identities
  
<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/b/b016/b016350/b0163507.png" /></td> <td valign="top" style="width:5%;text-align:right;">(*)</td></tr></table>
+
\begin{equation}x^2=J(xy,x,y)=0,\label{eq1}\end{equation}
  
 
where
 
where
  
<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/b/b016/b016350/b0163508.png" /></td> </tr></table>
+
$$J(x,y,z)=(xy)z+(yz)x+(zx)y.$$
  
If the characteristic of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016350/b0163509.png" /> is 2 and its cardinal number is not less than 4, the class of binary Lie algebras can be defined not only by the system of identities (*), but also by the identity
+
If the characteristic of $F$ is 2 and its cardinal number is not less than 4, the class of binary Lie algebras cannot be defined only by the system of identities \eqref{eq1}, also needed is 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/b/b016/b016350/b01635010.png" /></td> </tr></table>
+
$$J([(xy)y]x,x,y)=0.$$
  
The tangent algebra of an analytic local alternative [[Loop|loop]] is a binary Lie algebra and vice versa.
+
The tangent algebra of an analytic local alternative [[loop]] is a binary Lie algebra and vice versa.
  
 
====References====
 
====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A.I. Mal'tsev,   "Analytic loops"  ''Mat. Sb.'' , '''36 (78)''' :  3  (1955)  pp. 569–575  (In Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  A.T. Gainov,   "Binary Lie algebras of characteristic two"  ''Algebra and Logic'' , '''8''' :  5  (1969)  pp. 287–297  ''Algebra i Logika'' , '''8''' :  5  (1969)  pp. 505–522</TD></TR></table>
+
<table>
 +
<TR><TD valign="top">[1]</TD> <TD valign="top">  A.I. Mal'tsev, "Analytic loops"  ''Mat. Sb.'' , '''36 (78)''' :  3  (1955)  pp. 569–575  (In Russian)</TD></TR>
 +
<TR><TD valign="top">[2]</TD> <TD valign="top">  A.T. Gainov, "Binary Lie algebras of characteristic two"  ''Algebra and Logic'' , '''8''' :  5  (1969)  pp. 287–297  ''Algebra i Logika'' , '''8''' :  5  (1969)  pp. 505–522</TD></TR>
 +
</table>

Latest revision as of 06:57, 30 March 2024

$BL$-algebra

A linear algebra $A$ over a field $F$ any two elements of which generate a Lie subalgebra. The class of all binary Lie algebras over a given field $F$ generates a variety which, if the characteristic of $F$ is different from 2, is given by the system of identities

\begin{equation}x^2=J(xy,x,y)=0,\label{eq1}\end{equation}

where

$$J(x,y,z)=(xy)z+(yz)x+(zx)y.$$

If the characteristic of $F$ is 2 and its cardinal number is not less than 4, the class of binary Lie algebras cannot be defined only by the system of identities \eqref{eq1}, also needed is the identity

$$J([(xy)y]x,x,y)=0.$$

The tangent algebra of an analytic local alternative loop is a binary Lie algebra and vice versa.

References

[1] A.I. Mal'tsev, "Analytic loops" Mat. Sb. , 36 (78) : 3 (1955) pp. 569–575 (In Russian)
[2] A.T. Gainov, "Binary Lie algebras of characteristic two" Algebra and Logic , 8 : 5 (1969) pp. 287–297 Algebra i Logika , 8 : 5 (1969) pp. 505–522
How to Cite This Entry:
Binary Lie algebra. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Binary_Lie_algebra&oldid=18121
This article was adapted from an original article by A.T. Gainov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article