Quasitriangular Hopf algebra

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

$R$ for all $x$ , where $Δ$ is the coproduct on H, and the linear map $T :$ is given by $T(x$ ,

$(\Delta$ ,

$(1$ ,

where $R 12 = ϕ 12 (R)$ , $R 13 = ϕ 13 (R)$ , and $R 23 = ϕ 23 (R)$ , where $\phi_{12} :$ , $\phi_{13} :$ , and $\phi_{23} :$ , are algebra morphisms determined by

$\phi_{12}(a$

$\phi_{13}(a$

$\phi_{23}(a$

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, $(\epsilon$ ; moreover $R^{-1}$ , $R$ , and $(S$ . One may further show that the antipode S must be a linear isomorphism, and thus S^2 is an automorphism. In fact, S^2 is given by conjugating by an invertible element: $S (x) = u x u - 1$ where $u$ (cf. Ribbon Hopf algebras).

It is possible to construct a quasitriangular Hopf algebra from a Hopf algebra and its dual, using the Drinfel'd quantum double construction.

[edit] Twisting

The property of being a quasi-triangular Hopf algebra is preserved by twisting via an invertible element $F$ such that $(\varepsilon$ and satisfying the cocycle condition

$(F$

Furthermore,

u =	∑	fⁱS(f_i)
	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 Drinfel'd) twist.

[edit] See also

[edit] Notes

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

[edit] References

Susan Montgomery, Hans-Jürgen Schneider. New directions in Hopf algebras, Volume 43. Cambridge University Press, 2002. ISBN 9780521815123

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