Binary Lie algebra
From HandWiki
$BL$-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 |
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