A Hom-Lie algebra
has a bilinear skew-symmetric bracket and a linear map
. Considering a Hom-Lie algebra as an
-twisted version of a Lie algebra, in the paper under review the author studies the corresponding twisted Yang-Baxter equation: If
is a linear map of the vector space
, then the bilinear map
is a solution of the Hom-Yang-Baxter equation (HYBE) if
. The author shows that just as a Lie algebra gives a solution of the YBE, a Hom-Lie algebra gives a solution of the HYBE. He also constructs two other solutions of the HYBE from Drinfeld’s (dual) quasi-triangular bialgebras. Each solution of the HYBE can be extended to operators that satisfy the braid relations. Assuming an invertibility condition, these operators give a representation of the braid group.