Drinfeld Twist

In mathematics, a Hopf algebra, ''H'', is quasitriangularMontgomery & Schneider (2002), p. 72 if there exists an invertible element, ''R'', of $H \otimes H$ such that

:*$R \ \Delta(x)R^ = (T \circ \Delta)(x)$ 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 $T(x \otimes y) = y \otimes x$,

:*$(\Delta \otimes 1)(R) = R_ \ R_$,

:*$(1 \otimes \Delta)(R) = R_ \ R_$,

where $R_ = \phi_(R)$, $R_ = \phi_(R)$, and $R_ = \phi_(R)$, where $\phi_ : H \otimes H \to H \otimes H \otimes H$, $\phi_ : H \otimes H \to H \otimes H \otimes H$, and $\phi_ : H \otimes H \to H \otimes H \otimes H$, are algebra morphisms determined by

:$\phi_(a \otimes b) = a \otimes b \otimes 1,$

:$\phi_(a \otimes b) = a \otimes 1 \otimes b,$

:$\phi_(a \otimes b) = 1 \otimes a \otimes b.$

''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
$R^ = (S \otimes 1)(R)$, $R = (1 \otimes S)(R^)$, 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: $S^2(x)= u x u^$ where $u := m (S \otimes 1)R^$ (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
:$c_(u\otimes v) = T \left( R \cdot (u \otimes v )\right) = T \left( R_1 u \otimes R_2 v\right)$.

Twisting

The property of being a quasi-triangular Hopf algebra is preserved by twisting via an invertible element $F = \sum_i f^i \otimes f_i \in \mathcal$ 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 $S'(a) = u S(a)u^$, 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

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{{DEFAULTSORT:Quasitriangular Hopf Algebra
Hopf algebras