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Home > Hopf algebra

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(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.

1 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 function s 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.

2 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 " quantizationIn physics, quantization is the formulation of a classical theory in the formalism of quantum physics. Even though classical physics stems from quantum theory, the build up of a quantum theory is often made the other way around, starting from existing clas" 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 geometryIn mathematics, there is a close relationship between spaces which are geometric in nature, and the numerical functions on them. In general such functions will form a commutative ring, say the ring of C ''X of continuous functions on a topological space X, 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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Topics: Hopf Algebra Group K''''g Rarr Algebraic Delta Algebras...

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