Hopf algebra

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

${\displaystyle S(c_{(1)})c_{(2)}=c_{(1)}S(c_{(2)})=\varepsilon (c)1\qquad {\mbox{ for all }}c\in H.}$

As for algebras, one can replace the underlying field K with a commutative ring R in the above definition.[4]

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.[5]

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.

Let A be a Hopf algebra, and let M and N be A-modules. Then, M ⊗ N is also an A-module, with

${\displaystyle a(m\otimes n):=\Delta (a)(m\otimes n)=(a_{1}\otimes a_{2})(m\otimes n)=(a_{1}m\otimes a_{2}n)}$

for m ∈ M, n ∈ N and Δ(a) = (a₁, a₂). Furthermore, we can define the trivial representation as the base field K with

${\displaystyle a(m):=\epsilon (a)m}$

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

${\displaystyle (af)(m):=f(S(a)m)}$

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(m) is also a homomorphism of A-modules. However, the map M ⊗ M* → K is not necessarily a homomorphism of A-modules.

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 generalizations of Hopf algebras, where coassociativity only holds up to a twist. They have been used in the study of the Knizhnik–Zamolodchikov equations.[21]

Multiplier Hopf algebras introduced by Alfons Van Daele in 1994[22] are generalizations of Hopf algebras where comultiplication from an algebra (with or without unit) to the multiplier algebra of tensor product algebra of the algebra with itself.

Hopf group-(co)algebras introduced by V. G. Turaev in 2000 are also generalizations of Hopf algebras.

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".[25]

The definition of Hopf algebra is naturally extended to arbitrary braided monoidal categories.[26][27] A Hopf algebra in such a category ${\displaystyle (C,\otimes ,I,\alpha ,\lambda ,\rho ,\gamma )}$ is a sextuple ${\displaystyle (H,\nabla ,\eta ,\Delta ,\varepsilon ,S)}$ where ${\displaystyle H}$ is an object in ${\displaystyle C}$, and

${\displaystyle \nabla :H\otimes H\to H}$ (multiplication),

${\displaystyle \eta :I\to H}$ (unit),

${\displaystyle \Delta :H\to H\otimes H}$ (comultiplication),

${\displaystyle \varepsilon :H\to I}$ (counit),

${\displaystyle S:H\to H}$ (antipode)

— are morphisms in ${\displaystyle C}$ such that

1) the triple ${\displaystyle (H,\nabla ,\eta )}$ is a monoid in the monoidal category ${\displaystyle (C,\otimes ,I,\alpha ,\lambda ,\rho ,\gamma )}$, i.e. the following diagrams are commutative:[lower-alpha 2]

2) the triple ${\displaystyle (H,\Delta ,\varepsilon )}$ is a comonoid in the monoidal category ${\displaystyle (C,\otimes ,I,\alpha ,\lambda ,\rho ,\gamma )}$, i.e. the following diagrams are commutative:[lower-alpha 2]

3) the structures of monoid and comonoid on ${\displaystyle H}$ are compatible: the multiplication ${\displaystyle \nabla }$ and the unit ${\displaystyle \eta }$ are morphisms of comonoids, and (this is equivalent in this situation) at the same time the comultiplication ${\displaystyle \Delta }$ and the counit ${\displaystyle \varepsilon }$ are morphisms of monoids; this means that the following diagrams must be commutative:

where ${\displaystyle \lambda _{I}:I\otimes I\to I}$ is the left unit morphism in ${\displaystyle C}$, and ${\displaystyle \theta }$ the natural transformation of functors ${\displaystyle (A\otimes B)\otimes (C\otimes D){\stackrel {\theta }{\rightarrowtail }}(A\otimes C)\otimes (B\otimes D)}$ which is unique in the class of natural transformations of functors composed from the structural transformations (associativity, left and right units, transposition, and their inverses) in the category ${\displaystyle C}$.

The quintuple ${\displaystyle (H,\nabla ,\eta ,\Delta ,\varepsilon )}$ with the properties 1),2),3) is called a bialgebra in the category ${\displaystyle (C,\otimes ,I,\alpha ,\lambda ,\rho ,\gamma )}$;

4) the diagram of antipode is commutative:

The typical examples are the following.

• Groups. In the monoidal category ${\displaystyle ({\text{Set}},\times ,1)}$ of sets (with the cartesian product ${\displaystyle \times }$ as the tensor product, and an arbitrary singletone, say, ${\displaystyle 1=\{\varnothing \}}$, as the unit object) a triple ${\displaystyle (H,\nabla ,\eta )}$ is a monoid in the categorical sense if and only if it is a monoid in the usual algebraic sense, i.e. if the operations ${\displaystyle \nabla (x,y)=x\cdot y}$ and ${\displaystyle \eta (1)}$ behave like usual multiplication and unit in ${\displaystyle H}$ (but possibly without the invertibility of elements ${\displaystyle x\in H}$). At the same time, a triple ${\displaystyle (H,\Delta ,\varepsilon )}$ is a comonoid in the categorical sense iff ${\displaystyle \Delta }$ is the diagonal operation ${\displaystyle \Delta (x)=(x,x)}$ (and the operation ${\displaystyle \varepsilon }$ is defined uniquely as well: ${\displaystyle \varepsilon (x)=\varnothing }$). And any such a structure of comonoid ${\displaystyle (H,\Delta ,\varepsilon )}$ is compatible with any structure of monoid ${\displaystyle (H,\nabla ,\eta )}$ in the sense that the diagrams in the section 3 of the definition always commute. As a corollary, each monoid ${\displaystyle (H,\nabla ,\eta )}$ in ${\displaystyle ({\text{Set}},\times ,1)}$ can naturally be considered as a bialgebra ${\displaystyle (H,\nabla ,\eta ,\Delta ,\varepsilon )}$ in ${\displaystyle ({\text{Set}},\times ,1)}$, and vice versa. The existence of the antipode ${\displaystyle S:H\to H}$ for such a bialgebra ${\displaystyle (H,\nabla ,\eta ,\Delta ,\varepsilon )}$ means exactly that every element ${\displaystyle x\in H}$ has an inverse element ${\displaystyle x^{-1}\in H}$ with respect to the multiplication ${\displaystyle \nabla (x,y)=x\cdot y}$. Thus, in the category of sets ${\displaystyle ({\text{Set}},\times ,1)}$ Hopf algebras are exactly groups in the usual algebraic sense.
• Classical Hopf algebras. In the special case when ${\displaystyle (C,\otimes ,s,I)}$ is the category of vector spaces over a given field ${\displaystyle K}$, the Hopf algebras in ${\displaystyle (C,\otimes ,s,I)}$ are exactly the classical Hopf algebras described above.
• Functional algebras on groups. The standard functional algebras ${\displaystyle {\mathcal {C}}(G)}$, ${\displaystyle {\mathcal {E}}(G)}$, ${\displaystyle {\mathcal {O}}(G)}$, ${\displaystyle {\mathcal {P}}(G)}$ (of continuous, smooth, holomorphic, regular functions) on groups are Hopf algebras in the category (Ste,${\displaystyle \odot }$) of stereotype spaces,[28]
• Group algebras. The stereotype group algebras ${\displaystyle {\mathcal {C}}^{\star }(G)}$, ${\displaystyle {\mathcal {E}}^{\star }(G)}$, ${\displaystyle {\mathcal {O}}^{\star }(G)}$, ${\displaystyle {\mathcal {P}}^{\star }(G)}$ (of measures, distributions, analytic functionals and currents) on groups are Hopf algebras in the category (Ste,${\displaystyle \circledast }$) of stereotype spaces.[28] These Hopf algebras are used in the duality theories for non-commutative groups.[29]

• Quasitriangular Hopf algebra
• Algebra/set analogy
• Representation theory of Hopf algebras
• Ribbon Hopf algebra
• Superalgebra
• Supergroup
• Anyonic Lie algebra
• Sweedler's Hopf algebra
• Hopf algebra of permutations
• Milnor–Moore theorem