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Braided Hopf algebra

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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, particularly the Nichols algebra of a braided vectorspace in that category.

Contents

The notion should not be confused with quasitriangular Hopf algebra.

Definition

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 H H Y D if

  • ( R , , η ) is a unital associative algebra, where the multiplication map : R × R R and the unit η : k R are maps of Yetter–Drinfeld modules,
  • ( R , Δ , ε ) is a coassociative coalgebra with counit ε , and both Δ and ε are maps of Yetter–Drinfeld modules,
  • the maps Δ : R R R and ε : R k are algebra maps in the category H H Y D , where the algebra structure of R R is determined by the unit η η ( 1 ) : k R R and the multiplication map
  • Here c is the canonical braiding in the Yetter–Drinfeld category H H Y D .

    A braided bialgebra in H H Y D is called a braided Hopf algebra, if there is a morphism S : R R of Yetter–Drinfeld modules such that

    where Δ R ( r ) = r ( 1 ) r ( 2 ) in slightly modified Sweedler notation – a change of notation is performed in order to avoid confusion in Radford's biproduct below.

    Examples

  • Any Hopf algebra is also a braided Hopf algebra over H = k
  • A super Hopf algebra is nothing but a braided Hopf algebra over the group algebra H = k [ Z / 2 Z ] .
  • The tensor algebra T V of a Yetter–Drinfeld module V H H Y D is always a braided Hopf algebra. The coproduct Δ of T V is defined in such a way that the elements of V are primitive, that is
  • The counit ε : T V k then satisfies the equation ε ( v ) = 0 for all v V .
  • The universal quotient of T V , that is still a braided Hopf algebra containing V 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.
  • Radford's biproduct

    For any braided Hopf algebra R in H H Y D there exists a natural Hopf algebra R # H 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.

    As a vector space, R # H is just R H . The algebra structure of R # H is given by

    where r , r R , h , h H , Δ ( h ) = h ( 1 ) h ( 2 ) (Sweedler notation) is the coproduct of h H , and . : H R R is the left action of H on R. Further, the coproduct of R # H is determined by the formula

    Here Δ R ( r ) = r ( 1 ) r ( 2 ) denotes the coproduct of r in R, and δ ( r ( 2 ) ) = r ( 2 ) ( 1 ) r ( 2 ) ( 0 ) is the left coaction of H on r ( 2 ) R .

    References

    Braided Hopf algebra Wikipedia


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