Quasitriangular Hopf algebra

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In mathematics, a quasitriangular Hopf algebra is a particular kind of Hopf algebra ${\displaystyle H}$, equipped with a special element ${\displaystyle R\in H\otimes H}$, called the R-matrix. The R-matrix naturally provides the structure of a braiding on the rigid category of representations of ${\displaystyle H}$, which makes quasitriangular Hopf algebras useful in knot theory.

The most important examples of quasitriangular Hopf algebras are quantum groups.

Definition

A Hopf algebra H is quasitriangular if there exists an invertible element ${\displaystyle R\in H\otimes H}$ such that

• ${\displaystyle R\ \Delta (x)R^{-1}=(T\circ \Delta )(x)}$ for all ${\displaystyle x\in H}$, where ${\displaystyle \Delta }$ is the coproduct on H, and the linear map ${\displaystyle T:H\otimes H\to H\otimes H}$ is given by ${\displaystyle T(x\otimes y)=y\otimes x}$,

• ${\displaystyle (\Delta \otimes 1)(R)=R_{13}\ R_{23}}$,

• ${\displaystyle (1\otimes \Delta )(R)=R_{13}\ R_{12}}$,

where ${\displaystyle R_{12}=\phi _{12}(R)}$, ${\displaystyle R_{13}=\phi _{13}(R)}$, and ${\displaystyle R_{23}=\phi _{23}(R)}$, where ${\displaystyle \phi _{12}:H\otimes H\to H\otimes H\otimes H}$, ${\displaystyle \phi _{13}:H\otimes H\to H\otimes H\otimes H}$, and ${\displaystyle \phi _{23}:H\otimes H\to H\otimes H\otimes H}$, are algebra morphisms determined by

${\displaystyle \phi _{12}(a\otimes b)=a\otimes b\otimes 1,}$

${\displaystyle \phi _{13}(a\otimes b)=a\otimes 1\otimes b,}$

${\displaystyle \phi _{23}(a\otimes b)=1\otimes a\otimes b.}$

R is called the R-matrix.

Properties

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, ${\displaystyle (\epsilon \otimes 1)R=(1\otimes \epsilon )R=1\in H}$; moreover ${\displaystyle R^{-1}=(S\otimes 1)(R)}$, ${\displaystyle R=(1\otimes S)(R^{-1})}$, and ${\displaystyle (S\otimes S)(R)=R}$. 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: ${\displaystyle S^{2}(x)=uxu^{-1}}$ where ${\displaystyle u:=m((S\otimes 1)\circ T)R}$ (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

${\displaystyle c_{U,V}(u\otimes v)=T\left(R\cdot (u\otimes v)\right)=T\left(R_{1}u\otimes R_{2}v\right)}$.

Twisting

The property of being a quasi-triangular Hopf algebra is preserved by twisting via an invertible element ${\displaystyle F=\sum _{i}f^{i}\otimes f_{i}\in {\mathcal {A\otimes A}}}$ such that ${\displaystyle (\varepsilon \otimes id)F=(id\otimes \varepsilon )F=1}$ and satisfying the cocycle condition

${\displaystyle (F\otimes 1)\cdot (\Delta \otimes id)(F)=(1\otimes F)\cdot (id\otimes \Delta )(F)}$

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

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

Notes

References

• Montgomery, Susan (1993). Hopf algebras and their actions on rings. Regional Conference Series in Mathematics. Vol. 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. Vol. 43. Cambridge University Press. ISBN 978-0-521-81512-3. Zbl 0990.00022.