Difference between revisions of "Binary Lie algebra"
From Encyclopedia of Mathematics
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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 | 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 | where | ||
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$$J(x,y,z)=(xy)z+(yz)x+(zx)y.$$ | $$J(x,y,z)=(xy)z+(yz)x+(zx)y.$$ | ||
| − | 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 | + | 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 |
$$J([(xy)y]x,x,y)=0.$$ | $$J([(xy)y]x,x,y)=0.$$ | ||
| − | The tangent algebra of an analytic local alternative [[ | + | 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, | + | <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=43184
Binary Lie algebra. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Binary_Lie_algebra&oldid=43184
This article was adapted from an original article by A.T. Gainov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article