# Hopf algebra

In abstract algebra, a Hopf algebra is a bialgebra H over a field K together with a K-linear map ${\displaystyle S:H\rightarrow H}$ 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

${\displaystyle S(c_{(1)})c_{(2)}=c_{(1)}S(c_{(2)})=\epsilon (c)1\qquad {\mbox{ for all }}c\in C.}$

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 : KGKGKG by Δ(g) = gg for all g in G
• ε : KGK by ε(g) = 1 for all g in G
• S : KGKG 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 KGKG is naturally isomorphic to KGxG (for G infinite, KGKG is a proper subset of KGxG). The set KG becomes a Hopf algebra if we define

• Δ : KGKGxG by Δ(f)(x,y)=f(xy) for all f in KG and all x,y in G
• ε : KGK by ε(f) = f(e) for every f in KG [here e is the identity element of G]
• S : KGKG 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.

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 : UUU 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 : UK by ε(x) = 0 for all x in g (again, extended to U)
• S : UU 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 : HHHH is defined by T(xy) = yx). 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.