# Binary Lie algebra

From Encyclopedia of Mathematics

*-algebra*

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

(*) |

where

If the characteristic of 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

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