Hopf algebra




The notion of Hopf algebra is an abstraction of the properties of

where not only the associative algebra structure is remembered, but also the natural coalgebra structure, making it a bialgebra, as well as the algebraic structure induced by the inverse-operation in the group, called the antipode.

More intrinsically, a Hopf algebra structure on an associative algebra is precisely the structure such as to make its category of modules into a rigid monoidal category equipped with a fiber functor – this is the statement of Tannaka duality for Hopf algebras.

Hopf algebras and their generalization to Hopf algebroids arise notably as groupoid convolution algebras. Another important source of Hopf algebras is combinatorics, see at combinatorial Hopf algebras.

There is a wide variety of variations of the notion of Hopf algebra, relaxing properties or adding structure. Examples are weak Hopf algebras, quasi-Hopf algebras, (quasi-)triangular Hopf algebras, quantum groups, hopfish algebras etc. Most of these notions are systematized via Tannaka duality by the properties and structures on the coresponding categories of modules, see at Tannaka duality below.




A kk-bialgebra (A,m,η,Δ,ϵ)(A,m,\eta,\Delta,\epsilon) with multiplication mm, comultiplication Δ\Delta, unit η:kA\eta: k\to A and counit ϵ:Ak\epsilon \colon A\to k is called a Hopf algebra if there exists a kk-linear function

S:AA, S \colon A \longrightarrow A \,,

then called the antipode or coinverse, such that

m(idS)Δ=m(Sid)Δ=ηϵ m\circ(\mathrm{id}\otimes S)\circ \Delta \;=\; m\circ(S\otimes\mathrm{id})\circ\Delta = \eta\circ\epsilon

(as a map AAA\to A).


If the antipode of a Hopf algebra (Def. ) is an anti-involution, in that SS=idS \circ S = id, then one speaks of an involutive Hopf algebra.


If an antipode exists then it is unique, just the way that if inverses exist in a monoid then they are unique. One sometimes prefers to have a skew-antipode S˜\tilde{S} such that h (2)S˜(h (1))=S˜(h (2))h (1)=(ηϵ)(h)h_{(2)}\tilde{S}(h_{(1)}) = \tilde{S}(h_{(2)}) h_{(1)} = (\eta\circ\epsilon) (h). If SS is an invertible antipode then S˜S 1\tilde{S} - S^{-1} is a skew-antipode. HH is a bialgebra with a skew-antipode iff H opH^{op} (the same vector space, opposite product, the same coproduct) is a Hopf algebra.

The unit of a Hopf algebra is a grouplike element, hence S(1)1=1S(1)1=1, therefore S(1)=1S(1)=1. By linearity of SS this implies that Sηϵ=ηϵS\circ\eta\circ\epsilon = \eta\circ\epsilon.


The antipode is an anti-homomorphism both of algebras and coalgebras (i.e. a homomorphism S:AA op copS \colon A\to A^{cop}_{op}).

In particular, an involutive Hopf algebra (Def. ) is a star-algebra.

Proof (algebra part)

In Sweedler notation, for any g,hAg,h\in A,

S((hg) (1))(hg) (2)=ϵ(hg)S((h g)_{(1)}) (h g)_{(2)} = \epsilon(h g)
S((hg) (1))h (2)g (2)=ϵ(h)ϵ(g)S((h g)_{(1)}) h_{(2)}g_{(2)} = \epsilon(h)\epsilon(g)
S((hg) (1))h (2)g (2)Sg (3)Sh (3)=ϵ(h (1))ϵ(g (1))Sg (2)Sh (2)S((h g)_{(1)}) h_{(2)}g_{(2)} S g_{(3)} S h_{(3)} = \epsilon(h_{(1)})\epsilon(g_{(1)}) S g_{(2)} S h_{(2)}
S(h (1)g (1))ϵ(h (2))ϵ(g (2))=(Sg)(Sh)S(h_{(1)}g_{(1)}) \epsilon(h_{(2)})\epsilon(g_{(2)}) = (S g) (S h)
S(h (1)ϵ(h (2))g (1)ϵ(g (2)))=(Sg)(Sh)S(h_{(1)}\epsilon(h_{(2)})g_{(1)}\epsilon(g_{(2)})) = (S g)(S h)

Therefore S(hg)=(Sg)(Sh)S(h g) = (S g) (S h).

For the coalgebra part, notice first that ϵ(h)11=τΔ(ϵ(h)1)=τΔ(Sh (1)h (2))\epsilon(h)1\otimes 1 = \tau\circ\Delta(\epsilon(h)1)=\tau\circ\Delta(S h_{(1)}h_{(2)}). Expand this as

(Sh (1)Sh (2))(h (4)h (3))=((Sh (1)) (2)(Sh (1)) (1))(h (4)h (3)) (S h_{(1)}\otimes S h_{(2)})(h_{(4)}\otimes h_{(3)}) = ((S h_{(1)})_{(2)}\otimes (S h_{(1)})_{(1)})(h_{(4)}\otimes h_{(3)})
(Sh (1)Sh (2))(h (4)h (3))(Sh (5)Sh (6))=((Sh (1)) (2)(Sh (1)) (1))(h (3)h (2))(Sh (4)Sh (5)) (S h_{(1)}\otimes S h_{(2)})(h_{(4)}\otimes h_{(3)}) (S h_{(5)}\otimes S h_{(6)}) = ((S h_{(1)})_{(2)}\otimes (S h_{(1)})_{(1)})(h_{(3)}\otimes h_{(2)})(S h_{(4)}\otimes S h_{(5)})
((Sh (1)Sh (2))ϵ(h (3))=((Sh (1)) (2)(Sh (1)) (1))(1ϵ(h (2))) ((S h_{(1)}\otimes S h_{(2)})\epsilon(h_{(3)}) = ((S h_{(1)})_{(2)}\otimes (S h_{(1)})_{(1)})(1\otimes\epsilon(h_{(2)}))
((Sh (1)Sh (2))ϵ(h (3))=(τΔ)(Sh (1))(1ϵ(h (2))1)=(τΔ)(Sh (1)ϵ(h (2))) ((S h_{(1)}\otimes S h_{(2)})\epsilon(h_{(3)}) = (\tau\circ\Delta)(S h_{(1)})(1\otimes\epsilon(h_{(2)})1) = (\tau\circ\Delta)(S h_{(1)}\epsilon(h_{(2)}))
(Sh (1)Sh (2))=(τΔ)(Sh)=(Sh) (2)(Sh) (1). (S h_{(1)}\otimes S h_{(2)})=(\tau\circ\Delta)(S h) = (S h)_{(2)}\otimes (S h)_{(1)}.

The axiom that must be satisfied by the antipode looks like a kk-linear version of the identity satisfied by the inverse map in a group bimonoid: taking a group element gg, duplicating by the diagonal map Δ\Delta to obtain (g,g)(g,g), taking the inverse of either component of this ordered pair, and then multiplying the two components, we obtain the identity element of our group.

Just as an algebra is a monoid in Vect and a bialgebra is a bimonoid in VectVect, a Hopf algebra is a Hopf monoid in VectVect.


Caution: convention in topology

In algebraic topology, it is common to define Hopf algebras without mentioning the antipode, since in many topological cases of interest it exists automatically. For example, this is the case if it is graded and “connected” in the sense that its degree-0 part is just the ground field (a property possessed by the homology or cohomology of any connected space). In algebraic topology also the strict coassociativity is not always taken for granted.


Relation to Hopf groups

Note that the definition of Hopf algebra (or, really, of Hopf monoid) is self-dual: a Hopf monoid in a symmetric monoidal category VV is the same as a Hopf monoid in the opposite category V opV^{op} (i.e. a “Hopf comonoid”). Thus we can view a Hopf algebra as “like a group” in two different ways, depending on whether the group multiplication corresponds to the multiplication or the comultiplication of the Hopf algebra. The formal connections between Hopf monoids and group objects are:

  1. A Hopf monoid in a cartesian monoidal category VV is the same as a group object in VV. Such Hopf monoids are always cocommutative (that is, their underlying comonoid is cocommutative). This is because every object of a cartesian monoidal category is a cocommutative comonoid object in a unique way, and every morphism is a comonoid homomorphism.

  2. A commutative Hopf monoid in a symmetric monoidal category VV is the same as a group object in the opposite category CMon(V) opCMon(V)^{op}, where CMon(V)CMon(V) is the category of commutative monoids in VV, hence a cogroup object in CMon(V)CMon(V) (a point highlighted by Haynes Miller, see (Ravenel 86, appendix A1)). This works because the tensor product of commutative algebras is the categorical coproduct, and hence the product in its opposite category. In particular, a commutative Hopf algebra is the same as a group object in the category Alg opAlg^{op} of affine schemes.

Corresponding to these two, an ordinary group GG gives us two different Hopf algebras (here kk is the ground ring):

  1. The group algebra k[G]k[G] (the free vector space on the set GG), with multiplication given by the group operation of GG and comultiplication given by the diagonal gggg\mapsto g\otimes g. This Hopf algebra is always cocommutative, and is commutative iff GG is abelian. It can be viewed as the result of applying the strong monoidal functor k[]:SetkModk[-]:Set \to k Mod to the Hopf monoid GG in SetSet.

  2. The function algebra k(G)k(G) (the set of functions GkG\to k), with comultiplication given by precomposition with the group operation

    k(G)k(G×G)k(G)k(G),k(G) \to k(G\times G) \cong k(G)\otimes k(G),

    and multiplication given by pointwise multiplication in kk. In this case we need some finiteness or algebraicity of GG in order to guarantee k(G×G)k(G)k(G)k(G\times G) \cong k(G)\otimes k(G). This Hopf algebra is always commutative, and is cocommutative iff GG is abelian.

Note that if GG is finite, then k[G]k(G)k[G]\cong k(G) as kk-modules, but the Hopf algebra structure is quite different.

Mike, can you do something with these notes that I took at some point as a grad student? I don't know this stuff very well, which is why I don't incorporate them into the text, but at least I cleaned up the formatting a bit so that you can if you like it. —Toby

One can make a group into a Hopf algebra in at least 22 very different ways. Both ways have a discrete version and a smooth version.

Given a (finite, discrete) group GG and a ground ring (field?) KK, then the group ring K[G]K[G] is a cocommutative Hopf algebra, with M(g 0,g 1)=g 0g 1M(g_0,g_1) = g_0 g_1, I=1I = 1, D(g)=ggD(g) = g \otimes g, E(g)=1E(g) = 1, and the nifty Hopf antipodal operator S(g)=g 1S(g) = g^{-1}. Notice that the coalgebra operations D,ED,E depend only on Set|G|Set|G|.

Given a (finite, discrete) group GG and a ground ring (field?) KK, then the function ring Fun(G,K)Fun(G,K) is a commutative Hopf algebra, with M(f 0,f 1)(g)=f 0(g)f 1(g)M(f_0,f_1)(g) = f_0(g)f_1(g), I(g)=1I(g) = 1, D(f)(g,h)=f(gh)D(f)(g,h) = f(g h), E(f)=f(1)E(f) = f(1), and the nifty Hopf antipodal operator S(f)(g)=f(g 1)S(f)(g) = f(g^{-1}). Notice that the algebra operations M,IM,I depend only on Set|G|Set|G|.

Given a (simply connected) Lie group GG and the complex (real?) field KK, then the universal enveloping algebra U(G)U(G) is a cocommutative Hopf algebra, with M(g 0,g 1)=g 0g 1M(\mathbf{g}_0,\mathbf{g}_1) = \mathbf{g}_0 \mathbf{g}_1, I=1I = 1, D(g)=g1+1gD(\mathbf{g}) = \mathbf{g} \otimes 1 + 1 \otimes \mathbf{g}, E(g)=0E(\mathbf{g}) = 0, and the nifty Hopf antipodal operator S(g)=gS(\mathbf{g}) = -\mathbf{g}. Notice that the coalgebra operation D,ED,E depend only on KVect|𝔤|K Vect|\mathfrak{g}|.

Given a (compact) Lie group GG and the complex (real?) field KK, then the algebraic function ring Anal(G)Anal(G) is a cocommutative Hopf algebra, with M(f 0,f 1)(g)=f 0(g)f 1(g)M(f_0,f_1)(g) = f_0(g) f_1(g), I(g)=1I(g) = 1, D(f)(g,h)=f(gh)D(f)(g,h) = f(g h), E(f)=f(1)E(f) = f(1), and the nifty Hopf antipodal operator S(f)(g)=f(g 1)S(f)(g) = f(g^{-1}). Notice that the algebra operations M,IM,I depend only on AnalMan|G|Anal Man|G|.

The theorem of Hopf modules

Hopf algebras can be characterized among bialgebras by the fundamental theorem on Hopf modules: the category of Hopf modules over a bialgebra is canonically equivalent to the category of vector spaces over the ground ring iff the bialgebra is a Hopf algebra. This categorical fact enables a definition of Hopf monoids in some setups that do not allow a sensible definition of antipode.

Relation to Lie algebras

Tannaka duality

The category of modules (finite dimensional) over the underlying associative algebra of a Hopf algebra canonically inherits the structure of an rigid monoidal category such that the forgetful fiber functor to vector spaces over the ground field is a strict monoidal functor.

The statement of Tannaka duality for Hopf algebras is that this property characterizes Hopf algebras. (See for instance (Bakke))

For generalization of this characterization to quasi-Hopf algebras and hopfish algebras see (Vercruysse).

Tannaka duality for categories of modules over monoids/associative algebras

monoid/associative algebracategory of modules
AAMod AMod_A
RR-algebraMod RMod_R-2-module
sesquialgebra2-ring = monoidal presentable category with colimit-preserving tensor product
bialgebrastrict 2-ring: monoidal category with fiber functor
Hopf algebrarigid monoidal category with fiber functor
hopfish algebra (correct version)rigid monoidal category (without fiber functor)
weak Hopf algebrafusion category with generalized fiber functor
quasitriangular bialgebrabraided monoidal category with fiber functor
triangular bialgebrasymmetric monoidal category with fiber functor
quasitriangular Hopf algebra (quantum group)rigid braided monoidal category with fiber functor
triangular Hopf algebrarigid symmetric monoidal category with fiber functor
supercommutative Hopf algebra (supergroup)rigid symmetric monoidal category with fiber functor and Schur smallness
form Drinfeld doubleform Drinfeld center
trialgebraHopf monoidal category

2-Tannaka duality for module categories over monoidal categories

monoidal category2-category of module categories
AAMod AMod_A
RR-2-algebraMod RMod_R-3-module
Hopf monoidal categorymonoidal 2-category (with some duality and strictness structure)

3-Tannaka duality for module 2-categories over monoidal 2-categories

monoidal 2-category3-category of module 2-categories
AAMod AMod_A
RR-3-algebraMod RMod_R-4-module

As 3-vector spaces

A Hopf algebra structure on an associative algebra AA canonically defines on AA the structure of an algebra object internal to the 2-category of algebras, bimodules and bimodule homomorphisms.

By the discussion at n-vector space this allows to identify Hopf algebras with certain 3-vector spaces .

(For instance (FHLT, p. 27)).

More general 3-vector spaces are given by hopfish algebras and generally by sesquiunital sesquialgebras.



The diagrammatic definition of a Hopf algebra, is also in the Wikipedia entry.

  • Eiichi Abe, Hopf algebras, Cambridge UP 1980.

  • Pierre Cartier, A primer on Hopf algebras, Frontiers in number theory, physics, and geometry II, 537–615, preprint IHÉS 2006-40, 81p (doi:10.1007/978-3-540-30308-4_12, pdf, pdf, pdf)

  • V. G. Drinfel'd, Quantum groups, Proceedings of the International Congress of Mathematicians 1986, Vol. 1, 2 798–820, AMS 1987, djvu:1.3 M, pdf:2.5 M

  • G. Hochschild, Introduction to algebraic group schemes, 1971

  • Shahn Majid, Foundations of quantum group theory, Cambridge University Press 1995, 2000.

  • John Milnor, John Moore, On the structure of Hopf algebras, Annals of Math. 81 (1965), 211-264 (doi:10.2307/1970615, pdf)

  • Susan Montgomery, Hopf algebras and their action on rings, AMS 1994, 240p.

  • B. Parshall, J.Wang, Quantum linear groups, Mem. Amer. Math. Soc. 89(1991), No. 439, vi+157 pp.

  • Moss Sweedler, Hopf algebras, Benjamin 1969.

  • William C. Waterhouse, Introduction to affine group schemes, Graduate Texts in Mathematics 66, Springer 1979. xi+164 pp.

Tannaka duality for Hopf algebras and their generalization is alluded to in

  • Joost Vercruysse, Hopf algebras—Variant notions and reconstruction theorems (arXiv:1202.3613)

and discussed in detail in

  • Tørris Koløen Bakke, Hopf algebras and monoidal categories (2007) (pdf)

Discussion in algebraic topology with an eye towards stable homotopy theory, Steenrod algebra and the Adams spectral sequence:

Last revised on May 11, 2021 at 01:29:39. See the history of this page for a list of all contributions to it.