Difference between revisions of "Binary Lie algebra"
From Encyclopedia of Mathematics
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| + | ''$BL$-algebra'' | ||
| − | A linear 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 |
| − | + | $$x^2=J(xy,x,y)=0,\tag{*}$$ | |
where | where | ||
| − | + | $$J(x,y,z)=(xy)z+(yz)x+(zx)y.$$ | |
| − | If the characteristic of | + | If the characteristic of $F$ 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 \ref{*}, but also by the identity |
| − | + | $$J([(xy)y]x,x,y)=0.$$ | |
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|loop]] is a binary Lie algebra and vice versa. | ||
Revision as of 10:58, 9 November 2014
$BL$-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
$$x^2=J(xy,x,y)=0,\tag{*}$$
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 can be defined not only by the system of identities \ref{*}, but also by 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
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