# Lie ternary system

A vector space $\mathfrak m$ with a trilinear composition

$$\mathfrak m \times \mathfrak m \times \mathfrak m \rightarrow \mathfrak m ,\ \ ( X , Y , Z ) \rightarrow [ X , Y , Z ] ,$$

satisfying the following conditions:

$$[ X , X , Y ] = 0 ,$$

$$[ X , Y , Z ] + [ Y , Z , X ] + [ Z , X , Y ] = 0 ,$$

$$[ X , Y , [ Z , U , V ] ] =$$

$$= \ [ [ X , Y , Z ] , U , V ] + [ Z , [ X , Y , U ] , V ] + [ Z , U , [ X , Y , V ] ] .$$

If $\mathfrak g$ is a Lie algebra and $\mathfrak m \subset \mathfrak g$ is a subspace such that $[ [ X , Y ] , Z ] \in \mathfrak m$ for any $X , Y , Z \in \mathfrak m$, then the operation

$$[ X , Y , Z ] = [ [ X , Y ] , Z ]$$

converts $\mathfrak m$ into a Lie ternary system. Conversely, every Lie ternary system can be obtained in this way from some Lie algebra.

The category of finite-dimensional Lie ternary systems over the field $\mathbf R$ is equivalent to the category of simply-connected symmetric homogeneous spaces (see Symmetric space).

#### References

 [1] S. Helgason, "Differential geometry, Lie groups, and symmetric spaces" , Acad. Press (1978) [2] O. Loos, "Symmetric spaces" , 1 , Benjamin (1969)
How to Cite This Entry:
Lie ternary system. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Lie_ternary_system&oldid=47633
This article was adapted from an original article by A.S. Fedenko (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article