# Hopf algebra

In abstract algebra, a **Hopf algebra** is a bialgebra *H* over a field **K** together with a *K*-linear map such that the following diagram commutes

Here Δ is the comultiplication of the bialgebra, ∇ its multiplication, η its unit and ε its counit. In the sumless Sweedler notation, this property can also be expressed as

The map *S* is called the **antipode map** of the Hopf algebra.

## Contents

## Examples

**Group algebra.** Suppose *G* is a group. The group algebra *KG* is a unital associative algebra over *K*. It turns into a Hopf algebra if we define

- Δ :
*KG*→*KG*⊗*KG*by Δ(*g*) =*g*⊗*g*for all*g*in*G* - ε :
*KG*→*K*by ε(*g*) = 1 for all*g*in*G* *S*:*KG*→*KG*by*S*(*g*) =*g*^{ -1}for all*g*in*G*.

**Functions on a finite group.** Suppose now that *G* is a *finite* group. Then the set *K*^{G} of all functions from *G* to *K* with pointwise addition and multiplication is a unital associative algebra over *K*, and *K*^{G} ⊗ *K*^{G} is naturally isomorphic to *K*^{GxG} (for *G* infinite, *K*^{G} ⊗ *K*^{G} is a proper subset of *K*^{GxG}). The set *K*^{G} becomes a Hopf algebra if we define

- Δ :
*K*^{G}→*K*^{GxG}by Δ(*f*)(*x*,*y*)=*f*(*xy*) for all*f*in*K*^{G}and all*x*,*y*in*G* - ε :
*K*^{G}→*K*by ε(*f*) =*f*(*e*) for every*f*in*K*^{G}[here*e*is the identity element of*G*] *S*:*K*^{G}→*K*^{G}by*S*(*f*)(*x*) =*f*(*x*^{-1}) for all*f*in*K*^{G}and all*x*in*G*.

**Regular functions on an algebraic group.** Generalizing the previous example, we can use the same formulas to show that for a given algebraic group *G* over *K*, the set of all regular functions on *G* forms a Hopf algebra.

**Universal enveloping algebra.** Suppose *g* is a Lie algebra over the field *K* and *U* is its universal enveloping algebra. *U* becomes a Hopf algebra if we define

- Δ :
*U*→*U*⊗*U*by Δ(*x*) =*x*⊗1 + 1⊗*x*for every*x*in*g*(this rule is compatible with commutators and can therefore be uniquely extended to all of*U*). - ε :
*U*→*K*by ε(*x*) = 0 for all*x*in*g*(again, extended to*U*) *S*:*U*→*U*by*S*(*x*) = -*x*for all*x*in*g*.

## Quantum groups and non-commutative geometry

All examples above are either commutative (i.e. the multiplication is commutative) or co-commutative (i.e. Δ = Δ o *T* where *T* : *H*⊗*H* → *H*⊗*H* is defined by *T*(*x*⊗*y*) = *y*⊗*x*). The most exciting Hopf algebras however are certain "deformations" or "quantizations" of those from example 3 and 4 which are neither commutative nor co-commutative. These Hopf algebras are often called **quantum groups**, a term that is only loosely defined. They are important in noncommutative geometry, the idea being the following: a standard algebraic group is well described by its standard Hopf algebra of regular functions; we can then think of the deformed version of this Hopf algebra as describing a certain "non-standard" or "quantized" algebraic group (which is not an algebraic group at all). While there does not seem to be a direct way to define or manipulate these non-standard objects, one can still work with their Hopf algebras, and indeed one *identifies* them with their Hopf algebras. Hence the name "quantum group".

## Related concepts

Graded Hopf algebras are often used in algebraic topology: they are the natural algebraic structure of the totality of all homology or cohomology groups of a space.

Locally compact quantum groups generalize Hopf algebras and carry a topology. The algebra of all continuous functions on a Lie group is a locally compact quantum group.

## See also

## References

- Jurgen Fuchs,
*Affine Lie Algebras and Quantum Groups*, (1992), Cambridge University Press. ISBN 0-521-48412-X - Ross Moore, Sam Williams and Ross Talent:Quantum Groups: an entrée to modern algebra