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.
- 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 KG of all functions from G to K with pointwise addition and multiplication is a unital associative algebra over K, and KG ⊗ KG is naturally isomorphic to KGxG (for G infinite, KG ⊗ KG is a proper subset of KGxG). The set KG becomes a Hopf algebra if we define
- Δ : KG → KGxG by Δ(f)(x,y)=f(xy) for all f in KG and all x,y in G
- ε : KG → K by ε(f) = f(e) for every f in KG [here e is the identity element of G]
- S : KG → KG by S(f)(x) = f(x-1) for all f in KG 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.
- 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".
- 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