In mathematics, a Hopf algebra, named after Heinz Hopf, is a structure that is simultaneously a (unital associative) algebra and a (counital coassociative) coalgebra, with these structures' compatibility making it a bialgebra, and that moreover is equipped with an antihomomorphism satisfying a certain property.
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.
[3] Formally, a Hopf algebra is an (associative and coassociative) bialgebra H over a field K together with a K-linear map S: H → 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 As for algebras, one can replace the underlying field K with a commutative ring R in the above definition.
In general, S is an antihomomorphism,[6] so S2 is a homomorphism, which is therefore an automorphism if S was invertible (as may be required).
If H is finite-dimensional semisimple over a field of characteristic zero, commutative, or cocommutative, then it is involutive.
[7] 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.
The Nichols–Zoeller freeness theorem of Warren Nichols and Bettina Zoeller (1989) 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.
A Hopf subalgebra A is said to be right normal in a Hopf algebra H if it satisfies the condition of stability, adr(h)(A) ⊆ A for all h in H, where the right adjoint mapping adr is defined by adr(h)(a) = S(h(1))ah(2) 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(h)(a) = h(1)aS(h(2)).
A normal Hopf subalgebra A in H satisfies the condition (of equality of subsets of H): HA+ = A+H where A+ denotes the kernel of the counit on A.
[10] 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.
The group-like elements form a group with inverse given by the antipode.
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.
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.
Locally compact quantum groups generalize Hopf algebras and carry a topology.
The unit H-module is the separable algebra HL mentioned above.
In this case: In this philosophy, a group can be thought of as a Hopf algebra over the "field with one element".
[25] The definition of Hopf algebra is naturally extended to arbitrary braided monoidal categories.