Journal of the
Korean Mathematical Society JKMS

In this paper, we introduce the notion of generating index $\I_1(A)$ ($2$-generating index $\I_2(A)$, resp.) of a left-symmetric algebra $A$, which is the maximum of the dimensions of the subalgebras generated by any element (any two elements, resp.). We give a classification of left-symmetric algebras with $\I_1(A)=1$ and $\I_2(A)=2,3$ resp., and show that all such algebras can be constructed by linear and bilinear functions. Such algebras can be regarded as a generalization of those relating to the integrable (generalized) Burgers equation.

Keywords: left-symmetric algebra, generating index, non-associative algebra, linear function

MSC numbers: 17B05, 17B60, 17D99