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Hopf Algebra


Given a commutative ring R, an R-algebra H is a Hopf algebra if it has additional structure given by R-algebra homomorphisms

 Delta:H->H tensor _RH
(1)

(comultiplication) and

 epsilon:H->R
(2)

(counit) and an R-module homomorphism

 lambda:H->H
(3)

(antipode) that satisfy the properties

1. Coassociativity:

 (I tensor Delta)Delta=(Delta tensor I)Delta:H-->H tensor H tensor H.
(4)

2. Counitarity:

 m(I tensor epsilon)Delta=I=m(epsilon tensor I)Delta.
(5)

3. Antipode property:

 m(I tensor lambda)Delta=iotaepsilon=m(lambda tensor I)Delta,
(6)

where I is the identity map on H, m:H tensor H-->H is the multiplication in H, and iota:R->H is the R-algebra structure map for H, also called the unit map.

Faà di Bruno's formula can be cast in a framework that is a special case of a Hopf algebra (Figueroa and Gracia-Bondía 2005).

HopfAlgebra1

Coassociativity means that the above diagram commutes, meaning if the arrows were reversed and m were exchanged for Delta, a diagram illustrating the associativity of the multiplication within H would be obtained. And since iota embeds the ground ring R into H, the counit maps H into R. The counitarity property is similarly the dual of that satisfied by iota. With epsilon and Delta, an algebra becomes a bialgebra, but it is the addition of the antipode that makes H a Hopf algebra. The antipode should be thought of as an inverse on H similar to that which exists within a group, and the antipode is an anti-homomorphism at the level of algebras and co-algebras, meaning that

lambda(hh^')=lambda(h^')lambda(h)
(7)
Deltalambda(h)=(lambda tensor lambda)tauDelta,
(8)

where tau(h tensor h^')=h^' tensor h, which is called the switch map. Moreover, as with the inverse operation in a group, in many cases, the antipode is an involution.

The prototypical examples of Hopf algebras are group rings, where G is a finite group and H=R[G] is a Hopf algebra via

Delta(g)=g tensor g
(9)
epsilon(g)=1_R
(10)
lambda(g)=g^(-1)
(11)

for g in G and extend by linearity to all of R[G].

For general Hopf algebras, the comultiplication is given in Sweedler notation. That is, if h in H then

 Delta(h)=sum_((h))h_((1)) tensor h_((2)),
(12)

which allows by coassociativity

(I tensor Delta)Delta(h)=(Delta tensor I)Delta(h)
(13)
=sum_((h))h_((1)) tensor h_((2)) tensor h_((3)) in H tensor H tensor H
(14)

to be unambiguously written.

HopfAlgebra2

Hopf algebras can be categorized into different types by dualizing the distinctions one makes between algebras. For example, if H is commutative, this is equivalent to saying that m:H tensor H->H satisfies the property that m degreestau=m where tau is the switch map mentioned above. Likewise, a Hopf algebra is said to be cocommutative if tau degreesDelta=Delta, that is, if the above diagram commutes. Moreover, commutativity and cocommutativity are independent properties, and so Hopf algebras can be considered that satisfy one or the other, or both, or neither properties.

Additionally, just as the linear dual of an algebra is an algebra, the dual of a Hopf algebra H is also a Hopf algebra, where the algebra structure of H becomes the coalgebra structure of H^*, and vice-versa, and the antipode for H translates into an antipode for H^* in a canonical fashion.


See also

Faà di Bruno's Formula, Hopf Map

This entry contributed by Timothy Kohl

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References

Bergman, G. "Everybody Knows What a Hopf Algebra Is." Amer. Math. Soc. Contemp. Math 43, 25-48, 1985.Childs, L. Taming Wild Extensions: Hopf Algebras and Local Galois Module Theory. Providence, RI: Amer. Math. Soc., 2000.Figueroa, H. and Gracia-Bondía, J. M. "Combinatorial Hopf Algebras in Quantum Field Theory I." 19 Mar 2005. http://arxiv.org/abs/hep-th/0408145.Kassel, C. Quantum Groups. New York: Springer-Verlag, 1995.Montgomery, S. Hopf Algebras and Their Actions on Rings. Providence, RI: Amer. Math. Soc., 1993.Sweedler, M. E. Hopf Algebras. Reading, MA: Addison-Wesley, 1969.

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Hopf Algebra

Cite this as:

Kohl, Timothy. "Hopf Algebra." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/HopfAlgebra.html

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