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The Hom-Yang-Baxter equation, Hom-Lie algebras, and quasi-triangular bialgebras. (English) Zbl 1179.17001
A Hom-Lie algebra $L$ has a bilinear skew-symmetric bracket and a linear map $\alpha:L\to L$ such that $\alpha([x,y])=[\alpha(x),\alpha(y)]$ and $[[x,y],\alpha(z)]+[[z,x],\alpha(y)]+[[y,z],\alpha(x)]=0$, $x,y,z\in L$. Considering a Hom-Lie algebra as an $\alpha$-twisted version of a Lie algebra, in the paper under review the author studies the corresponding twisted Yang-Baxter equation: If $\alpha:M\to M$ is a linear map of the vector space $M$, then the bilinear map $B:M\otimes M\to M\otimes M$ is a solution of the Hom-Yang-Baxter equation (HYBE) if $B\circ \alpha^{\otimes 2}=\alpha^{\otimes 2}\circ B$ and $(\alpha\otimes B)\circ (B\otimes \alpha)\circ (\alpha\otimes B) =(B\otimes \alpha)\circ (\alpha\otimes B)\circ (B\otimes \alpha)$. 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.

##### MSC:
 17A30 Nonassociative algebras satisfying other identities 16T05 Hopf algebras and their applications 17B37 Quantum groups and related deformations 81R50 Quantum groups and related algebraic methods in quantum theory
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