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Quasitriangular Hopf algebra
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<p>In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, a <a href="/facts/Hopf_algebra/3QF6g0tT">Hopf algebra</a>, <i>H</i>, is quasitriangular if <a href="/facts/There_exists/hHveaBYo">there exists</a> an <a href="/facts/Inverse_element/CwMygHSj">invertible</a> element, <i>R</i>, of H ⊗ H {\displaystyle H\otimes H} such that </p> <ul><li> R Δ ( x ) R − 1 = ( T ∘ Δ ) ( x ) {\displaystyle R\ \Delta (x)R^{-1}=(T\circ \Delta )(x)} for all x ∈ H {\displaystyle x\in H} , where Δ {\displaystyle \Delta } is the coproduct on <i>H</i>, and the linear map T : H ⊗ H → H ⊗ H {\displaystyle T:H\otimes H\to H\otimes H} is given by T ( x ⊗ y ) = y ⊗ x {\displaystyle T(x\otimes y)=y\otimes x} ,</li></ul> <ul><li> ( Δ ⊗ 1 ) ( R ) = R 13 R 23 {\displaystyle (\Delta \otimes 1)(R)=R_{13}\ R_{23}} ,</li></ul> <ul><li> ( 1 ⊗ Δ ) ( R ) = R 13 R 12 {\displaystyle (1\otimes \Delta )(R)=R_{13}\ R_{12}} ,</li></ul> <p>where R 12 = ϕ 12 ( R ) {\displaystyle R_{12}=\phi _{12}(R)} , R 13 = ϕ 13 ( R ) {\displaystyle R_{13}=\phi _{13}(R)} , and R 23 = ϕ 23 ( R ) {\displaystyle R_{23}=\phi _{23}(R)} , where ϕ 12 : H ⊗ H → H ⊗ H ⊗ H {\displaystyle \phi _{12}:H\otimes H\to H\otimes H\otimes H} , ϕ 13 : H ⊗ H → H ⊗ H ⊗ H {\displaystyle \phi _{13}:H\otimes H\to H\otimes H\otimes H} , and ϕ 23 : H ⊗ H → H ⊗ H ⊗ H {\displaystyle \phi _{23}:H\otimes H\to H\otimes H\otimes H} , are algebra <a href="/facts/Morphism/Kdpuyz3S">morphisms</a> determined by </p> ϕ 12 ( a ⊗ b ) = a ⊗ b ⊗ 1 , {\displaystyle \phi _{12}(a\otimes b)=a\otimes b\otimes 1,} ϕ 13 ( a ⊗ b ) = a ⊗ 1 ⊗ b , {\displaystyle \phi _{13}(a\otimes b)=a\otimes 1\otimes b,} ϕ 23 ( a ⊗ b ) = 1 ⊗ a ⊗ b . {\displaystyle \phi _{23}(a\otimes b)=1\otimes a\otimes b.} <p><i>R</i> is called the <a href="/facts/R-matrix/InCb8KVv">R-matrix</a>. </p><p>As a consequence of the properties of quasitriangularity, the R-matrix, <i>R</i>, is a solution of the <a href="/facts/Yang%25E2%2580%2593Baxter_equation/FEzKGDYh">Yang–Baxter equation</a> (and so a <a href="/facts/Module_(mathematics)/jP0LIhFL">module</a> <i>V</i> of <i>H</i> can be used to determine quasi-invariants of <a href="/facts/Braid_theory/rn5323Kx">braids</a>, <a href="/facts/Knot_(mathematics)/UMtKJ36m">knots</a> and <a href="/facts/Link_(knot_theory)/hoFDYmVj">links</a>). Also as a consequence of the properties of quasitriangularity, ( ϵ ⊗ 1 ) R = ( 1 ⊗ ϵ ) R = 1 ∈ H {\displaystyle (\epsilon \otimes 1)R=(1\otimes \epsilon )R=1\in H} ; moreover R − 1 = ( S ⊗ 1 ) ( R ) {\displaystyle R^{-1}=(S\otimes 1)(R)} , R = ( 1 ⊗ S ) ( R − 1 ) {\displaystyle R=(1\otimes S)(R^{-1})} , and ( S ⊗ S ) ( R ) = R {\displaystyle (S\otimes S)(R)=R} . One may further show that the antipode <i>S</i> must be a linear isomorphism, and thus <i>S2</i> is an automorphism. In fact, <i>S2</i> is given by conjugating by an invertible element: S 2 ( x ) = u x u − 1 {\displaystyle S^{2}(x)=uxu^{-1}} where u := m ( S ⊗ 1 ) R 21 {\displaystyle u:=m(S\otimes 1)R^{21}} (cf. <a href="/facts/Ribbon_Hopf_algebra/DctC2Y8w">Ribbon Hopf algebras</a>). </p><p>It is possible to construct a quasitriangular Hopf algebra from a Hopf algebra and its dual, using the <a href="/facts/Vladimir_Drinfeld/xGhJ8NAv">Drinfeld</a> quantum double construction. </p><p>If the Hopf algebra <i>H</i> is quasitriangular, then the category of modules over <i>H</i> is braided with braiding </p> c U , V ( u ⊗ v ) = T ( R ⋅ ( u ⊗ v ) ) = T ( R 1 u ⊗ R 2 v ) {\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)} .