Hopf algebra

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In mathematics, a Hopf algebra, named after Heinz Hopf, is a certain type of bialgebra structure, that is, simultaneously a (unital associative) algebra and a coalgebra (with these structures compatible). The extra datum in the definition of a Hopf algebra is the antipode map, which generalises the inversion map on a group that sends g to g − 1. Hopf algebras occur naturally in algebraic topology, where they originated and are related to the H-space concept, in group scheme theory, and in numerous other places, making them probably the most familiar type of bialgebra. Hopf algebras are also studied in their own right, with much work on specific classes of examples on the one hand and classification problems on the other.

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[edit] Formal definition

More formally, a Hopf algebra is a bialgebra H over a field K together with a K-linear map S:H\rightarrow H such that the following diagram commutes

antipode commutative diagram

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 S2 = Id, 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 S2 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.

[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 KG of all functions from G to K with pointwise addition and multiplication is a unital associative algebra over K, and KG ⊗ KG is naturally isomorphic to KGxG (for G infinite, KG ⊗ 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 ⊗ 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] Cohomology of Lie groups

The cohomology algebra of a Lie group is a Hopf algebra: the multiplication is provided by the cup-product, and the comultiplication

H^*(G) \mapsto H^*(G\times G) \cong H^*(G)\otimes H^*(G)

by the group multiplication G\times G\mapsto G. This observation was actually a source of the notion of Hopf algebra. Using this structure, Hopf proved a structure theorem for the cohomology algebra of Lie groups.

Theorem (Hopf) [1] Let A be a finite-dimensional, graded commutative, graded cocommutative Hopf algebra over a field of characteristic 0. Then A (as an algebra) is a free exterior algebra with generators of odd degree.

[edit] Quantum groups and non-commutative geometry

Main article: quantum group

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 so far 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] Notes

[edit] References

  1. ^ H. Hopf, Uber die Topologie der Gruppen-Mannigfaltigkeiten und ihrer Verallgemeinerungen, Ann. of Math. 42 (1941), 22-52. Reprinted in Selecta Heinz Hopf, pp. 119-151, Springer, Berlin (1964). MR4784
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