A linear Lie rack structure on a finite dimensional vector space V is a Lie rack operation pointed at the origin and such that for any x, the left translation is linear. A linear Lie rack operation is called analytic if for any where is an n + 1-multilinear map symmetric in the n first arguments. In this case, is exactly the left Leibniz product associated to Any left Leibniz algebra has a canonical analytic linear Lie rack structure given by where In this paper, we show that a sequence of n + 1-multilinear maps on a vector space V defines an analytic linear Lie rack structure if and only if is a left Leibniz bracket, the are invariant for and satisfy a sequence of multilinear equations. Some of these equations have a cohomological interpretation and can be solved when the zero and the 1-cohomology of the left Leibniz algebra are trivial. On the other hand, given a left Leibniz algebra we show that there is a large class of (analytic) linear Lie rack structures on which can be built from the canonical one and invariant multilinear symmetric maps on A left Leibniz algebra on which all the analytic linear Lie rack structures are build in this way will be called rigid. We use our characterizations of analytic linear Lie rack structures to show that and are rigid. We conjecture that any simple Lie algebra is rigid as a left Leibniz algebra.
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