Hopf algebra

In mathematics, a Hopf algebra, named after Heinz Hopf, is a structure that is simultaneously a (unital associative) algebra, a coalgebra, and has an antiautomorphism, with these structures compatible.

Hopf algebras occur naturally in algebraic topology, where they originated and are related to the H-space concept, in group scheme theory, in group theory (via the concept of a group ring), 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.

Formal definition

Formally, a Hopf algebra is a bialgebra H over a field K together with a K-linear map $S\colon H\to H$ (called the antipode) 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 H.

As for algebras, one can replace the underlying field K with a commutative ring R in the above definition.

The definition of Hopf algebra is self-dual (as reflected in the symmetry of the above diagram), so if one can define a dual of H (which is always possible if H is finite-dimensional), then it is automatically a Hopf algebra.

Properties of the antipode

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).

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).

If $S^{2}=Id$ , then the Hopf algebra is said to be involutive (and the underlying algebra with involution is a *-algebra). If H is finite-dimensional semisimple over a field of characteristic zero, commutative, or cocommutative, then it is involutive.

If a bialgebra B admits an antipode S, then S is unique ("a bialgebra admits at most 1 Hopf algebra structure").

The antipode is an analog to the inversion map on a group that sends $g$ to $g^{-1}$ .^[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 (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.

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)\rightarrow H^{*}(G\times G)\cong H^{*}(G)\otimes H^{*}(G)

by the group multiplication $G\times G\rightarrow G$ . 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)^[2] 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.

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 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".

Related concepts

Graded Hopf algebras are often used in algebraic topology: they are the natural algebraic structure on the direct sum of all homology or cohomology groups of an H-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.

Analogy with groups

Groups can be axiomatized by the same diagrams (equivalently, operations) as a Hopf algebra, where G is taken to be a set instead of a module. In this case:

the field K is replaced by the 1-point set
there is a natural counit (map to 1 point)
there is a natural comultiplication (the diagonal map)
the unit is the identity element of the group
the multiplication is the multiplication in the group
the antipode is the inverse

In this philosophy, a group can be thought of as a Hopf algebra over the "field with one element".^[3]

Notes

^ Quantum groups lecture notes
^ Hopf, 1941.
^ Group = Hopf algebra « Secret Blogging Seminar, Group objects and Hopf algebras, video of Simon Willerton.

References

Pierre Cartier, A primer of Hopf algebras, IHES preprint, September 2006, 81 pages
Jurgen Fuchs, Affine Lie Algebras and Quantum Groups, (1992), Cambridge University Press. ISBN 0-521-48412-X
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). MR 4784
Ross Moore, Sam Williams and Ross Talent: Quantum Groups: an entrée to modern algebra

[1] Quantum groups lecture notes

[2] Hopf, 1941.

[3] Group = Hopf algebra « Secret Blogging Seminar, Group objects and Hopf algebras, video of Simon Willerton.

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