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 $S:H\rightarrow H$ 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

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

The map S is called the antipode map of the Hopf algebra. If an antipode exists, it must be unique. S is sometimes required to have a K-linear inverse, which is automatic in the finite-dimensional case, or if H is commutative or cocommutative (or more generally quasitriangular). If $S 2 = I d$ , then the Hopf algebra is said to be involutive, which is always true if H is commutative or cocommutative. In general, S is an antihomomorphism, so $S 2$ is a homomorphism, which is therefore an automorphism if S was invertible (as may be required). The definition of Hopf algebra is self-dual, so if one can define a dual of H (which is always possible of H is finite-dimensional), then it is automatically a Hopf algebra.

1 Examples
2 Quantum groups and non-commutative geometry
3 Related concepts
4 See also
5 References

[edit] 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.

[edit] 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 $\circ$ Δ where T: H ⊗ H → H ⊗ H is defined by T(x ⊗ y) = y ⊗ 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".

[edit] 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.

Quasi-Hopf algebras are also generalizations of Hopf algebras, where coassociativity only holds up to a twist.

[edit] See also

[edit] 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

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Categories: Hopf algebras | Monoidal categories | Representation theory