Hopf algebra


In mathematics, a Hopf algebra, named after Heinz Hopf, is a structure that is simultaneously an algebra and a coalgebra, with these structures' compatibility making it a bialgebra, and that moreover is equipped with an antiautomorphism satisfying a certain property. The representation theory of a Hopf algebra is particularly nice, since the existence of compatible comultiplication, counit, and antipode allows for the construction of tensor products of representations, trivial representations, and dual representations.
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, 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. They have diverse applications ranging from condensed-matter physics and quantum field theory to string theory and LHC phenomenology.

Formal definition

Formally, a Hopf algebra is a bialgebra H over a field K together with a K-linear map S: H → 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
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, so if one can define a dual of H, then it is automatically a Hopf algebra.

Structure constants

Fixing a basis for the underlying vector space, one may define the algebra in terms of structure constants for multiplication:
for co-multiplication:
and the antipode:
Associativity then requires that
while co-associativity requires that
The connecting axiom requires that

Properties of the antipode

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.
In general, S is an antihomomorphism, so S2 is a homomorphism, which is therefore an automorphism if S was invertible.
If S2 = idH, then the Hopf algebra is said to be involutive. 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. Thus, the antipode does not pose any extra structure which we can choose: Being a Hopf algebra is a property of a bialgebra.
The antipode is an analog to the inversion map on a group that sends g to g−1.

Hopf subalgebras

A subalgebra A of a Hopf algebra H is a Hopf subalgebra if it is a subcoalgebra of H and the antipode S maps A into A. In other words, a Hopf subalgebra A is a Hopf algebra in its own right when the multiplication, comultiplication, counit and antipode of H is restricted to A. The Nichols–Zoeller freeness theorem established that the natural A-module H is free of finite rank if H is finite-dimensional: a generalization of Lagrange's theorem for subgroups. As a corollary of this and integral theory, a Hopf subalgebra of a semisimple finite-dimensional Hopf algebra is automatically semisimple.
A Hopf subalgebra A is said to be right normal in a Hopf algebra H if it satisfies the condition of stability, adr ⊆ A for all h in H, where the right adjoint mapping adr is defined by adr = Sah for all a in A, h in H. Similarly, a Hopf subalgebra A is left normal in H if it is stable under the left adjoint mapping defined by adl = haS. The two conditions of normality are equivalent if the antipode S is bijective, in which case A is said to be a normal Hopf subalgebra.
A normal Hopf subalgebra A in H satisfies the condition : HA+ = A+H where A+ denotes the kernel of the counit on K. This normality condition implies that HA+ is a Hopf ideal of H. As a consequence one has a quotient Hopf algebra H/HA+ and epimorphism H → H/A+H, a theory analogous to that of normal subgroups and quotient groups in group theory.

Hopf orders

A Hopf order O over an integral domain R with field of fractions K is an order in a Hopf algebra H over K which is closed under the algebra and coalgebra operations: in particular, the comultiplication Δ maps O to O⊗O.

Group-like elements

A group-like element is a nonzero element x such that Δ = x⊗x. The group-like elements form a group with inverse given by the antipode. A primitive element x satisfies Δ = x⊗1 + 1⊗x.

Examples

Note that functions on a finite group can be identified with the group ring, though these are more naturally thought of as dual – the group ring consists of finite sums of elements, and thus pairs with functions on the group by evaluating the function on the summed elements.

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
by the group multiplication. 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 Let be a finite-dimensional, graded commutative, graded cocommutative Hopf algebra over a field of characteristic 0. Then is a free exterior algebra with generators of odd degree.

Quantum groups and non-commutative geometry

All examples above are either commutative or co-commutative. 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. 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".

Representation theory

Let A be a Hopf algebra, and let M and N be A-modules. Then, M ⊗ N is also an A-module, with
for m ∈ M, n ∈ N and Δ =. Furthermore, we can define the trivial representation as the base field K with
for m ∈ K. Finally, the dual representation of A can be defined: if M is an A-module and M* is its dual space, then
where f ∈ M* and m ∈ M.
The relationship between Δ, ε, and S ensure that certain natural homomorphisms of vector spaces are indeed homomorphisms of A-modules. For instance, the natural isomorphisms of vector spaces M → M ⊗ K and M → K ⊗ M are also isomorphisms of A-modules. Also, the map of vector spaces M* ⊗ M → K with f ⊗ m → f is also a homomorphism of A-modules. However, the map M ⊗ M* → K is not necessarily a homomorphism of A-modules.

Related concepts

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 generalizations of Hopf algebras, where coassociativity only holds up to a twist. They have been used in the study of the Knizhnik–Zamolodchikov equations.
Multiplier Hopf algebras introduced by Alfons Van Daele in 1994 are generalizations of Hopf algebras where comultiplication from an algebra to the multiplier algebra of tensor product algebra of the algebra with itself.
Hopf group-algebras introduced by V. G. Turaev in 2000 are also generalizations of Hopf algebras.

Weak Hopf algebras

s, or quantum groupoids, are generalizations of Hopf algebras. Like Hopf algebras, weak Hopf algebras form a self-dual class of algebras; i.e., if H is a Hopf algebra, so is H*, the dual space of linear forms on H. A weak Hopf algebra H is usually taken to be a
  1. for all a in H or εs;
  2. for all a in H or εt;
  3. for all a in H.
The axioms are partly chosen so that the category of H-modules is a rigid monoidal category. The unit H-module is the separable algebra HL mentioned above.
For example, a finite groupoid algebra is a weak Hopf algebra. In particular, the groupoid algebra on with one pair of invertible arrows eij and eji between i and j in is isomorphic to the algebra H of n x n matrices. The weak Hopf algebra structure on this particular H is given by coproduct Δ = eij ⊗ eij, counit ε = 1 and antipode S = eji. The separable subalgebras HL and HR coincide and are non-central commutative algebras in this particular case.
Early theoretical contributions to weak Hopf algebras are to be found in as well as

Hopf algebroids

See Hopf algebroid

Analogy with groups

Groups can be axiomatized by the same diagrams as a Hopf algebra, where G is taken to be a set instead of a module. In this case:
In this philosophy, a group can be thought of as a Hopf algebra over the "field with one element".

Hopf algebras in braided monoidal categories

The definition of Hopf algebra is naturally extended to arbitrary braided monoidal categories. A Hopf algebra in such a category is a sextuple where is an object in, and
— are morphisms in such that












The typical examples are the following.