Quasitriangular Hopf algebra explained

In mathematics, a Hopf algebra, H, is quasitriangular^[1] if there exists an invertible element, R, of

H ⊗ H such thatR \Delta(x)R ^{-1} = (T \circ \Delta)(x) for all x \in H , where \Delta is the coproduct on H, and the linear map T : H ⊗ H \to H ⊗ H is given by T(x ⊗ y) = y ⊗ x ,(\Delta ⊗ 1 )(R) = R _{13} R _{23} ,( 1 ⊗ \Delta)(R) = R _{13} R _{12} ,

where

R _{12} = \phi _{12} (R) , R _{13} = \phi _{13} (R) , and R _{23} = \phi _{23} (R) , where \phi _{12} : H ⊗ H \to H ⊗ H ⊗ H , \phi _{13} : H ⊗ H \to H ⊗ H ⊗ H , and \phi _{23} : H ⊗ H \to H ⊗ H ⊗ H , are algebra morphisms determined by\phi _{12} (a ⊗ b) = a ⊗ b ⊗ 1 , \phi _{13} (a ⊗ b) = a ⊗ 1 ⊗ b, \phi _{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,

(\epsilon ⊗ 1 ) R = ( 1 ⊗ \epsilon) R = 1 \in H ; moreover R ^{-1} = (S ⊗ 1 )(R) , R = ( 1 ⊗ S)(R ^{-1} ) , and (S ⊗ 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} 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.

If the Hopf algebra H is quasitriangular, then the category of modules over H is braided with braiding

c _{U,V} (u ⊗ v) = T \left( R ⋅ (u ⊗ v )\right) = T \left( R _{1} u ⊗ R _{2} v\right) .

Twisting

The property of being a quasi-triangular Hopf algebra is preserved by twisting via an invertible element

F = \sum _{i} f ^{i} ⊗ f _{i} \in l{A ⊗ A} such that (\varepsilon ⊗ id )F = (id ⊗ \varepsilon)F = 1 and satisfying the cocycle condition(F ⊗ 1 ) ⋅ (\Delta ⊗ id)( F) = ( 1 ⊗ F) ⋅ (id ⊗ \Delta)( F)

Furthermore,

u = \sum _{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.

See also

• Quasi-triangular quasi-Hopf algebra
• Ribbon Hopf algebra

Notes

1. Montgomery & Schneider (2002), [{{Google books|plainurl=y|id=I3IK9U5Co_0C|page=72|text=Quasitriangular}} p. 72].

References

• Book: Montgomery, Susan . Susan Montgomery

. Susan Montgomery . Hopf algebras and their actions on rings . Regional Conference Series in Mathematics . 82 . Providence, RI . . 1993 . 0-8218-0738-2 . 0793.16029 .

• Book: Montgomery, Susan . Susan Montgomery

. Susan Montgomery . Hans-Jürgen Schneider . Hans-Jürgen . Schneider . New directions in Hopf algebras . Mathematical Sciences Research Institute Publications . 43 . Cambridge University Press . 2002 . 978-0-521-81512-3 . 0990.00022 .