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.

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. 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^G 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

• Algebra/set analogy
• representation of a Hopf algebra
• superalgebra
• supergroup
• anyonic Lie algebra

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