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Quasitriangular Hopf algebra

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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.

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=(1\otimes \epsilon )R=1\in H} ; moreover R − 1 = ( S ⊗ 1 ) ( R ) {\displaystyle R^{-1}=(S\otimes 1)(R)} , R = ( 1 ⊗ S ) ( R − 1 ) {\displaystyle R=(1\otimes S)(R^{-1})} , and ( S ⊗ S ) ( R ) = R {\displaystyle (S\otimes S)(R)=R} . One may further show that the antipode S must be a linear isomorphism, and thus S is an automorphism. In fact, S 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(S\otimes 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 ( R ⋅ ( u ⊗ v ) ) = T ( R 1 u ⊗ R 2 v ) {\displaystyle c_{U,V}(u\otimes v)=T\left(R\cdot (u\otimes v)\right)=T\left(R_{1}u\otimes R_{2}v\right)} .

Twisting

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 {A\otimes A}}} such that ( ε ⊗ i d ) F = ( i d ⊗ ε ) F = 1 {\displaystyle (\varepsilon \otimes id)F=(id\otimes \varepsilon )F=1} and satisfying the cocycle condition

( F ⊗ 1 ) ⋅ ( Δ ⊗ i d ) ( F ) = ( 1 ⊗ F ) ⋅ ( i d ⊗ Δ ) ( F ) {\displaystyle (F\otimes 1)\cdot (\Delta \otimes id)(F)=(1\otimes F)\cdot (id\otimes \Delta )(F)}

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.

See also

Notes

  1. Montgomery & Schneider (2002), p. 72.

References

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