Quasitriangular Hopf algebra

In mathematics, a Hopf algebra, H, is quasitriangular if there exists an invertible element, R, of H ⊗ H {\displaystyle H\otimes H} such that R Δ ( x ) R − 1 = ( T ∘ Δ ) ( x ) {\displaystyle R\ \Delta (x)R^{-1}=(T\circ \Delta )(x)} for all x ∈ H {\displaystyle x\in H} , where Δ {\displaystyle \Delta } is the coproduct on H, and the linear map T : H ⊗ H → H ⊗ H {\displaystyle T:H\otimes H\to H\otimes H} is given by T ( x ⊗ y ) = y ⊗ x {\displaystyle T(x\otimes y)=y\otimes x} , ( Δ ⊗ 1 ) ( R ) = R 13 R 23 {\displaystyle (\Delta \otimes 1)(R)=R_{13}\ R_{23}} , ( 1 ⊗ Δ ) ( R ) = R 13 R 12 {\displaystyle (1\otimes \Delta )(R)=R_{13}\ R_{12}} , where R 12 = ϕ 12 ( R ) {\displaystyle R_{12}=\phi _{12}(R)} , R 13 = ϕ 13 ( R ) {\displaystyle R_{13}=\phi _{13}(R)} , and R 23 = ϕ 23 ( R ) {\displaystyle R_{23}=\phi _{23}(R)} , where ϕ 12 : H ⊗ H → H ⊗ H ⊗ H {\displaystyle \phi _{12}:H\otimes H\to H\otimes H\otimes H} , ϕ 13 : H ⊗ H → H ⊗ H ⊗ H {\displaystyle \phi _{13}:H\otimes H\to H\otimes H\otimes H} , and ϕ 23 : H ⊗ H → H ⊗ H ⊗ H {\displaystyle \phi _{23}:H\otimes H\to H\otimes H\otimes H} , are algebra morphisms determined by ϕ 12 ( a ⊗ b ) = a ⊗ b ⊗ 1 , {\displaystyle \phi _{12}(a\otimes b)=a\otimes b\otimes 1,} ϕ 13 ( a ⊗ b ) = a ⊗ 1 ⊗ b , {\displaystyle \phi _{13}(a\otimes b)=a\otimes 1\otimes b,} ϕ 23 ( a ⊗ b ) = 1 ⊗ a ⊗ b . {\displaystyle \phi _{23}(a\otimes b)=1\otimes a\otimes b.} R is called the R-matrix.

Source: Wikipedia — Quasitriangular Hopf algebra (CC BY-SA 4.0)