In mathematics, a Hopf algebra, H, is quasitriangular if there exists an invertible element, R, of H ⊗ H such that
R Δ ( x ) R − 1 = ( T ∘ Δ ) ( x ) for all x ∈ H , where Δ is the coproduct on H, and the linear map T : H ⊗ H → H ⊗ H is given by T ( x ⊗ y ) = y ⊗ x , ( Δ ⊗ 1 ) ( R ) = R 13 R 23 , ( 1 ⊗ Δ ) ( R ) = R 13 R 12 ,where R 12 = ϕ 12 ( R ) , R 13 = ϕ 13 ( R ) , and R 23 = ϕ 23 ( R ) , where ϕ 12 : H ⊗ H → H ⊗ H ⊗ H , ϕ 13 : H ⊗ H → H ⊗ H ⊗ H , and ϕ 23 : H ⊗ H → H ⊗ H ⊗ H , are algebra morphisms determined by
ϕ 12 ( a ⊗ b ) = a ⊗ b ⊗ 1 , ϕ 13 ( a ⊗ b ) = a ⊗ 1 ⊗ b , ϕ 23 ( a ⊗ b ) = 1 ⊗ a ⊗ b . 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 ; moreover R − 1 = ( S ⊗ 1 ) ( R ) , R = ( 1 ⊗ S ) ( R − 1 ) , and ( S ⊗ S ) ( R ) = R . One may further show that the antipode 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 where u := m ( S ⊗ 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.
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 such that ( ε ⊗ i d ) F = ( i d ⊗ ε ) F = 1 and satisfying the cocycle condition
( F ⊗ 1 ) ∘ ( Δ ⊗ i d ) F = ( 1 ⊗ F ) ∘ ( i d ⊗ Δ ) F Furthermore, u = ∑ i f i S ( f i ) is invertible and the twisted antipode is given by S ′ ( a ) = u S ( 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.
