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| In [[mathematics]], a '''braided Hopf algebra''' is a [[Hopf algebra]] in a [[braided monoidal category]]. The most common braided Hopf algebras are objects in a [[Yetter–Drinfeld category]] of a Hopf algebra ''H'', particurlarly the [[Nichols algebra]] of a braided vectorspace in that category.
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| ''The notion should not be confused with [[quasitriangular Hopf algebra]].''
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| == Definition ==
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| Let ''H'' be a Hopf algebra over a field ''k'', and assume that the antipode of ''H'' is bijective. A [[Yetter–Drinfeld module]] ''R'' over ''H'' is called a '''braided bialgebra''' in the Yetter–Drinfeld category <math> {}^H_H\mathcal{YD}</math> if
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| * <math> (R,\cdot ,\eta ) </math> is a unital [[associative algebra]], where the multiplication map <math>\cdot :R\times R\to R</math> and the unit <math> \eta :k\to R </math> are maps of Yetter–Drinfeld modules,
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| * <math> (R,\Delta ,\varepsilon )</math> is a coassociative [[coalgebra]] with counit <math>\varepsilon </math>, and both <math> \Delta </math> and <math>\varepsilon </math> are maps of Yetter–Drinfeld modules,
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| * the maps <math>\Delta :R\to R\otimes R </math> and <math> \varepsilon :R\to k </math> are algebra maps in the category <math> {}^H_H\mathcal{YD}</math>, where the algebra structure of <math> R\otimes R </math> is determined by the unit <math> \eta \otimes \eta(1) : k\to R\otimes R</math> and the multiplication map
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| :: <math> (R\otimes R)\times (R\otimes R)\to R\otimes R,\quad (r\otimes s,t\otimes u) \mapsto \sum _i rt_i\otimes s_i u, \quad \text{and}\quad c(s\otimes t)=\sum _i t_i\otimes s_i. </math>
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| :Here ''c'' is the canonical braiding in the Yetter–Drinfeld category <math> {}^H_H\mathcal{YD}</math>.
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| A braided bialgebra in <math> {}^H_H\mathcal{YD}</math> is called a '''braided Hopf algebra''', if there is a morphism <math> S:R\to R </math> of Yetter–Drinfeld modules such that
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| :: <math> S(r^{(1)})r^{(2)}=r^{(1)}S(r^{(2)})=\eta(\varepsilon (r)) </math> for all <math> r\in R,</math>
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| where <math>\Delta _R(r)=r^{(1)}\otimes r^{(2)}</math> in slightly modified [[Coalgebra|Sweedler notation]] – a change of notation is performed in order to avoid confusion in Radford's biproduct below.
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| == Examples ==
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| * Any Hopf algebra is also a braided Hopf algebra over <math> H=k </math>
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| * A '''super Hopf algebra''' is nothing but a braided Hopf algebra over the [[group ring|group algebra]] <math> H=k[\mathbb{Z}/2\mathbb{Z}] </math>.
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| * The [[tensor algebra]] <math> TV </math> of a Yetter–Drinfeld module <math> V\in {}^H_H\mathcal{YD}</math> is always a braided Hopf algebra. The coproduct <math> \Delta </math> of <math> TV </math> is defined in such a way that the elements of ''V'' are primitive, that is
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| ::<math> \Delta (v)=1\otimes v+v\otimes 1 \quad \text{for all}\quad v\in V.</math>
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| :The counit <math>\varepsilon :TV\to k</math> then satisfies the equation <math> \varepsilon (v)=0</math> for all <math> v\in V .</math>
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| * The universal quotient of <math> TV </math>, that is still a braided Hopf algebra containing <math> V </math> as primitive elements is called the [[Nichols algebra]]. They take the role of quantum Borel algebras in the classification of pointed Hopf algebras, analogously to the classical Lie algebra case.
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| == Radford's biproduct ==
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| For any braided Hopf algebra ''R'' in <math> {}^H_H\mathcal{YD}</math> there exists a natural Hopf algebra <math> R\# H </math> which contains ''R'' as a subalgebra and ''H'' as a Hopf subalgebra. It is called '''Radford's biproduct''', named after its discoverer, the Hopf algebraist David Radford. It was rediscovered by [[Shahn Majid]], who called it '''bosonization'''.
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| As a vector space, <math> R\# H </math> is just <math> R\otimes H </math>. The algebra structure of <math> R\# H </math> is given by
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| :: <math> (r\# h)(r'\#h')=r(h_{(1)}\boldsymbol{.}r')\#h_{(2)}h', </math>
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| where <math> r,r'\in R,\quad h,h'\in H</math>, <math> \Delta (h)=h_{(1)}\otimes h_{(2)} </math> ([[coproduct|Sweedler notation]]) is the coproduct of <math> h\in H </math>, and <math> \boldsymbol{.}:H\otimes R\to R </math> is the left action of ''H'' on ''R''. Further, the coproduct of <math> R\# H </math> is determined by the formula
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| :: <math> \Delta (r\#h)=(r^{(1)}\#r^{(2)}{}_{(-1)}h_{(1)})\otimes (r^{(2)}{}_{(0)}\#h_{(2)}), \quad r\in R,h\in H.</math>
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| Here <math>\Delta _R(r)=r^{(1)}\otimes r^{(2)}</math> denotes the coproduct of ''r'' in ''R'', and <math> \delta (r^{(2)})=r^{(2)}{}_{(-1)}\otimes r^{(2)}{}_{(0)} </math> is the left coaction of ''H'' on <math> r^{(2)}\in R. </math>
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| == References ==
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| * Andruskiewitsch, Nicolás and Schneider, Hans-Jürgen, ''Pointed Hopf algebras'', New directions in Hopf algebras, 1–68, Math. Sci. Res. Inst. Publ., 43, Cambridge Univ. Press, Cambridge, 2002.
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| [[Category:Hopf algebras]]
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