In mathematics, a Hopf algebra, named after Heinz Hopf, is a structure that is simultaneously an (unital associative) algebra and a (counital coassociative) 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.
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.
Fixing a basis for the underlying vector space, one may define the algebra in terms of structure constants for multiplication:
If S2 = idH, 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"). 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.
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 (and additionally the identity 1 of H is required to be in A). The Nichols–Zoeller freeness theorem established (in 1989) 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(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)). 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 (of equality of subsets of H): 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 (i.e. an algebra ideal in the kernel of the counit, a coalgebra coideal and stable under the antipode). 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.
A Hopf orderO over an integral domainR with field of fractionsK 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.
A group-like element is a nonzero element x such that Δ(x) = x⊗x. The group-like elements form a group with inverse given by the antipode. A primitive elementx satisfies Δ(x) = x⊗1 + 1⊗x.
in terms of complete homogeneous symmetric functions hk (k ≥ 1):
Δ(hk) = 1 ⊗ hk + h1 ⊗ hk−1 + ... + hk−1 ⊗ h1 + hk ⊗ 1.
ε(hk) = 0
S(hk) = (−1)kek
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 (over a field ) 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 (Hopf) Let be a finite-dimensional, graded commutative, graded cocommutative Hopf algebra over a field of characteristic 0. Then (as an algebra) is a free exterior algebra with generators of odd degree.
All examples above are either commutative (i.e. the multiplication is commutative) or co-commutative (i.e. Δ = T ∘ Δ where the twist mapT: 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".
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(a) = (a1, a2). 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(m) is also a homomorphism of A-modules. However, the map M ⊗ M* → K is not necessarily a homomorphism of A-modules.
Weak Hopf algebras, 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 (weak) Hopf algebra, so is H*, the dual space of linear forms on H (with respect to the algebra-coalgebra structure obtained from the natural pairing with H and its coalgebra-algebra structure). A weak Hopf algebra H is usually taken to be a
finite-dimensional algebra and coalgebra with coproduct Δ: H → H ⊗ H and counit ε: H → k satisfying all the axioms of Hopf algebra except possibly Δ(1) ≠ 1 ⊗ 1 or ε(ab) ≠ ε(a)ε(b) for some a,b in H. Instead one requires the following:
for all a, b, and c in H.
H has a weakened antipode S: H → H satisfying the axioms:
for all a in H (the right-hand side is the interesting projection usually denoted by ΠR(a) or εs(a) with image a separable subalgebra denoted by HR or Hs);
for all a in H (another interesting projection usually denoted by ΠR(a) or εt(a) with image a separable algebra HL or Ht, anti-isomorphic to HL via S);
for all a in H.
Note that if Δ(1) = 1 ⊗ 1, these conditions reduce to the two usual conditions on the antipode of a Hopf algebra.
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 [n] with one pair of invertible arrows eij and eji between i and j in [n] 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 ⊗ eij, counit ε(eij) = 1 and antipode S(eij) = eji. The separable subalgebras HL and HR coincide and are non-central commutative algebras in this particular case (the subalgebra of diagonal matrices).
Early theoretical contributions to weak Hopf algebras are to be found in as well as
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
1) the triple is a monoid in the monoidal category , i.e. the following diagrams are commutative:
2) the triple is a comonoid in the monoidal category , i.e. the following diagrams are commutative:
3) the structures of monoid and comonoid on are compatible: the multiplication and the unit are morphisms of comonoids, and (this is equivalent in this situation) at the same time the comultiplication and the counit are morphisms of monoids; this means that the following diagrams must be commutative:
the quintuple with the properties 1),2),3) is called a bialgebra in the category ;
4) the diagram of antipode is commutative:
The typical examples are the following.
Groups. In the monoidal category of sets (with the cartesian product as the tensor product, and an arbitrary singletone, say, , as the unit object) a triple 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 and behave like usual multiplication and unit in (but possibly without the invertibility of elements ). At the same time, a triple is a comonoid in the categorical sense iff is the diagonal operation (and the operation is defined uniquely as well: ). And any such a structure of comonoid is compatible with any structure of monoid in the sense that the diagrams in the section 3 of the definition always commute. As a corollary, each monoid in can naturally be considered as a bialgebra in , and vice versa. The existence of the antipode for such a bialgebra means exactly that every element has an inverse element with respect to the multiplication . Thus, in the category of sets Hopf algebras are exactly groups in the usual algebraic sense.
Classical Hopf algebras. In the special case when is the category of vector spaces over a given field , the Hopf algebras in are exactly the classical Hopf algebras described above.
Functional algebras on groups. The standard functional algebras, , , (of continuous, smooth, holomorphic, regular functions) on groups are Hopf algebras in the category (Ste,) of stereotype spaces,
^The finiteness of G implies that KG ⊗ KG is naturally isomorphic to KGxG. This is used in the above formula for the comultiplication. For infinite groups G, KG ⊗ KG is a proper subset of KGxG. In this case the space of functions with finite support can be endowed with a Hopf algebra structure.
^Hochschild, G (1965), Structure of Lie groups, Holden-Day, pp. 14–32
^ abHere , , are the natural transformations of associativity, and of the left and the right units in the monoidal category .
^Here is the left unit morphism in , and the natural transformation of functors 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 .
Fuchs, Jürgen (1992), Affine Lie algebras and quantum groups. An introduction with applications in conformal field theory, Cambridge Monographs on Mathematical Physics, Cambridge: Cambridge University Press, ISBN978-0-521-48412-1, Zbl0925.17031
Akbarov, S.S. (2003). "Pontryagin duality in the theory of topological vector spaces and in topological algebra". Journal of Mathematical Sciences. 113 (2): 179–349. doi:10.1023/A:1020929201133. S2CID115297067.