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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 α:LL such that α([x,y])=[α(x),α(y)] and [[x,y],α(z)]+[[z,x],α(y)]+[[y,z],α(x)]=0, x,y,zL. Considering a Hom-Lie algebra as an α-twisted version of a Lie algebra, in the paper under review the author studies the corresponding twisted Yang-Baxter equation: If α:MM is a linear map of the vector space M, then the bilinear map B:MMMM is a solution of the Hom-Yang-Baxter equation (HYBE) if Bα 2 =α 2 B and (αB)(Bα)(αB)=(Bα)(αB)(Bα). 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:
17A30Nonassociative algebras satisfying other identities
16T05Hopf algebras and their applications
17B37Quantum groups and related deformations
81R50Quantum groups and related algebraic methods in quantum theory