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A $G$-grading on an algebra is called multiplicity free if each homogeneouscomponent of the grading is 1-dimensional, where $G$ is an abelian group. Weintroduce skew root systems of Lie type and skew root systems of Jordan typerespectively, and use them to construct multiplicity free gradings onsemisimple Lie algebras and on semisimple Jordan algebras respectively. Undercertain conditions the corresponding Lie (resp. Jordan) algebras are simple.Three families of skew root systems of Lie type (resp. of Jordan type) areconstructed and the corresponding Lie (resp. Jordan) algebras are identified.This is a new approach to study abelian group gradings on Lie and Jordanalgebras.

Multiplicity free gradings on semisimple Lie and Jordan algebras and skew root