Hopf algebra

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In mathematics, a Hopf algebra, named after Heinz Hopf, is a bialgebra H over a field K together with a K-linear map <math>S:H\rightarrow H</math> 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

<math>S(c_{(1)})c_{(2)}=c_{(1)}S(c_{(2)})=\epsilon(c)1\qquad\mbox{ for all }c\in C.</math>

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

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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 Template:Dirprod KG by Δ(g) = g Template:Dirprod 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 Template:Dirprod KG is naturally isomorphic to KGxG (for G infinite, KG Template:Dirprod 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.

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 Template:Dirprod U by Δ(x) = x Template:Dirprod 1 + 1 Template:Dirprod 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.
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Quantum groups and non-commutative geometry

All examples above are either commutative (i.e. the multiplication is commutative) or co-commutative (i.e. Δ = T <math>\circ</math> Δ where T: H Template:Dirprod H → H Template:Dirprod H is defined by T(x Template:Dirprod y) = y Template:Dirprod x). Other interesting Hopf algebras are certain "deformations" or "quantizations" of those from example 3 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".

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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.

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See also

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References

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