Quasitriangular Hopf algebra

In mathematics, a Hopf algebra, H, is quasitriangular if there exists an invertible element, R, of H ⊗ H {displaystyle Hotimes H} such that In mathematics, a Hopf algebra, H, is quasitriangular if there exists an invertible element, R, of H ⊗ H {displaystyle Hotimes H} such that 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}:Hotimes H o Hotimes Hotimes H} , ϕ 13 : H ⊗ H → H ⊗ H ⊗ H {displaystyle phi _{13}:Hotimes H o Hotimes Hotimes H} , and ϕ 23 : H ⊗ H → H ⊗ H ⊗ H {displaystyle phi _{23}:Hotimes H o Hotimes Hotimes H} , are algebra morphisms determined by R is called the R-matrix. As a consequence of the properties of quasitriangularity, the R-matrix, R, is a solution of the Yang-Baxter equation (and so a module V of H can be used to determine quasi-invariants of braids, knots and links). Also as a consequence of the properties of quasitriangularity, ( ϵ ⊗ 1 ) R = ( 1 ⊗ ϵ ) R = 1 ∈ H {displaystyle (epsilon otimes 1)R=(1otimes epsilon )R=1in H} ; moreover R − 1 = ( S ⊗ 1 ) ( R ) {displaystyle R^{-1}=(Sotimes 1)(R)} , R = ( 1 ⊗ S ) ( R − 1 ) {displaystyle R=(1otimes S)(R^{-1})} , and ( S ⊗ S ) ( R ) = R {displaystyle (Sotimes S)(R)=R} . One may further show that theantipode S must be a linear isomorphism, and thus S2 is an automorphism. In fact, S2 is given by conjugating by an invertible element: S 2 ( x ) = u x u − 1 {displaystyle S^{2}(x)=uxu^{-1}} where u := m ( S ⊗ 1 ) R 21 {displaystyle u:=m(Sotimes 1)R^{21}} (cf. Ribbon Hopf algebras). It is possible to construct a quasitriangular Hopf algebra from a Hopf algebra and its dual, using the Drinfeld quantum double construction. If the Hopf algebra H is quasitriangular, then the category of modules over H is braided with braiding The property of being a quasi-triangular Hopf algebra is preserved by twisting via an invertible element F = ∑ i f i ⊗ f i ∈ A ⊗ A {displaystyle F=sum _{i}f^{i}otimes f_{i}in {mathcal {Aotimes A}}} such that ( ε ⊗ i d ) F = ( i d ⊗ ε ) F = 1 {displaystyle (varepsilon otimes id)F=(idotimes varepsilon )F=1} and satisfying the cocycle condition Furthermore, u = ∑ i f i S ( f i ) {displaystyle u=sum _{i}f^{i}S(f_{i})} is invertible and the twisted antipode is given by S ′ ( a ) = u S ( a ) u − 1 {displaystyle S'(a)=uS(a)u^{-1}} , with the twisted comultiplication, R-matrix and co-unit change according to those defined for the quasi-triangular quasi-Hopf algebra. Such a twist is known as an admissible (or Drinfeld) twist.