Quasitriangular Hopf algebra

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

• ${\displaystyle R\ \Delta (x)R^{-1}=(T\circ \Delta )(x)}$ 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: $S^{2}(x)=uxu^{{-1}}$ where $u:=m(S\otimes 1)R^{{21}}$ (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.

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

$(F\otimes 1)\circ (\Delta \otimes id)F=(1\otimes F)\circ (id\otimes \Delta )F$

Furthermore, $u=\sum _{i}f^{i}S(f_{i})$ is invertible and the twisted antipode is given by $S'(a)=uS(a)u^{{-1}}$, 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

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

Notes

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

References

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