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

  • R \ \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 \otimes H \to H \otimes H is given by T(x \otimes y) = y \otimes x,
  • (\Delta \otimes 1)(R) = R_{13} \ R_{23},
  • (1 \otimes \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 \otimes H \to H \otimes H \otimes H, \phi_{13} : H \otimes H \to H \otimes H \otimes H, and \phi_{23} : H \otimes H \to H \otimes H \otimes H, are algebra morphisms determined by

\phi_{12}(a \otimes b) = a \otimes b \otimes 1,
\phi_{13}(a \otimes b) = a \otimes 1 \otimes b,
\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, (\epsilon \otimes 1) R = (1 \otimes \epsilon) R = 1 \in H; moreover R^{-1} = (S \otimes 1)(R), R = (1 \otimes S)(R^{-1}), and (S \otimes 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 \otimes 1) \circ T)R (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\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 = \sum_i f^i \otimes f_i \in \mathcal{A \otimes A} such that (\varepsilon \otimes id )F = (id \otimes \varepsilon)F = 1 and satisfying the cocycle condition

(F \otimes 1) \cdot (\Delta \otimes id)( F) = (1 \otimes F) \cdot (id \otimes \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

Notes

  1. ^ Montgomery & Schneider (2002), [ p. 72].

References

  • Montgomery, Susan (1993). Hopf algebras and their actions on rings. Vol. 82. Regional Conference Series in Mathematics. Providence, RI: American Mathematical Society. ISBN 0-8218-0738-2. Zbl 0793.16029.
  • Montgomery, Susan & Schneider, Hans-Jürgen (2002). New directions in Hopf algebras. Vol. 43. Mathematical Sciences Research Institute Publications. Cambridge University Press. ISBN 978-0-521-81512-3. Zbl 0990.00022.