Quasitriangular Hopf algebra: Difference between revisions
m Spelling of "Drinfel'd" and addition of apostrophe to later occurance |
|||
Line 19: | Line 19: | ||
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 [[braid theory|braids]], [[knot (mathematics)|knots]] and [[link (knot theory)|links]]). Also as a consequence of the properties of quasitriangularity, <math>R^{-1} = (S \otimes 1)(R)</math>, and <math>R = (1 \otimes S)(R^{-1})</math>, and so <math>(S \otimes S)(R) = R</math>. |
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 [[braid theory|braids]], [[knot (mathematics)|knots]] and [[link (knot theory)|links]]). Also as a consequence of the properties of quasitriangularity, <math>R^{-1} = (S \otimes 1)(R)</math>, and <math>R = (1 \otimes S)(R^{-1})</math>, and so <math>(S \otimes S)(R) = R</math>. |
||
It is possible to contruct a quasitriangular Hopf algebra from a Hopf algebra and its dual, using the |
It is possible to contruct a quasitriangular Hopf algebra from a Hopf algebra and its dual, using the Drinfel'd [[quantum double]] construction. |
||
== Twisting == |
== Twisting == |
||
Line 26: | Line 26: | ||
:<math> (F \otimes 1) \circ (\Delta \otimes id) F = (1 \otimes F) \circ (id \otimes \Delta) F </math> |
:<math> (F \otimes 1) \circ (\Delta \otimes id) F = (1 \otimes F) \circ (id \otimes \Delta) F </math> |
||
Furthermore, <math> u = \sum_i f^i S(f_i)</math> is invertible and the twisted antipode is given by <math>S'(a) = u S(a)u^{-1}</math>, 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 |
Furthermore, <math> u = \sum_i f^i S(f_i)</math> is invertible and the twisted antipode is given by <math>S'(a) = u S(a)u^{-1}</math>, 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. |
||
== See Also == |
== See Also == |
Revision as of 14:37, 4 September 2006
In mathematics, a Hopf algebra, H, is quasitriangular if there exists an invertible element, R, of such that
- for all , where is the coproduct on H, and the linear map is given by ,
- ,
- ,
where , , and , where , , and , are algebra morphisms determined by
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, , and , and so .
It is possible to contruct a quasitriangular Hopf algebra from a Hopf algebra and its dual, using the Drinfel'd quantum double construction.
Twisting
The property of being a quasi-triangular Hopf algebra is preserved by twisting via an invertible element such that and satisfying the cocycle condition
Furthermore, is invertible and the twisted antipode is given by , 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.