nLab Hopf algebra

Idea

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

• the group algebra of a group;

• the algebra of functions on a finite group, and more generally, the algebra of regular functions on an affine algebraic $k$-group;

• the universal enveloping algebra of a Lie algebra,

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.

Definition

Definition

(antipode)

A $k$-bialgebra $(A,m,\eta,\Delta,\epsilon)$ with multiplication $m$, comultiplication $\Delta$, unit $\eta \colon k\to A$ and counit $\epsilon \colon A\to k$ is called a Hopf algebra if there exists a $k$-linear function

$$S \colon A \longrightarrow A \,,$$

then called the antipode or coinverse, such that

$$m\circ(\mathrm{id}\otimes S)\circ \Delta \;=\; m\circ(S\otimes\mathrm{id})\circ\Delta = \eta\circ\epsilon$$

(as a map $A\to A$).

Definition

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

Remark

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 $\tilde{S}$ such that $h_{(2)}\tilde{S}(h_{(1)}) = \tilde{S}(h_{(2)}) h_{(1)} = (\eta\circ\epsilon) (h)$. If $S$ is an invertible antipode then $\tilde{S} - S^{-1}$ is a skew-antipode. $H$ is a bialgebra with a skew-antipode iff $H^{op}$ (the same vector space, opposite product, the same coproduct) is a Hopf algebra.

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

Proposition

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

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

Proof (algebra part)

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

$$S((h g)_{(1)}) (h g)_{(2)} = \epsilon(h g)$$

$$S((h g)_{(1)}) h_{(2)}g_{(2)} = \epsilon(h)\epsilon(g)$$

$$S((h g)_{(1)}) h_{(2)}g_{(2)} S g_{(3)} S h_{(3)} = \epsilon(h_{(1)})\epsilon(g_{(1)}) S g_{(2)} S h_{(2)}$$

$$S(h_{(1)}g_{(1)}) \epsilon(h_{(2)})\epsilon(g_{(2)}) = (S g) (S h)$$

$$S(h_{(1)}\epsilon(h_{(2)})g_{(1)}\epsilon(g_{(2)})) = (S g)(S h)$$

Therefore $S(h g) = (S g) (S h)$.

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

$$(S h_{(1)}\otimes S h_{(2)})(h_{(4)}\otimes h_{(3)}) = ((S h_{(1)})_{(2)}\otimes (S h_{(1)})_{(1)})(h_{(4)}\otimes h_{(3)})$$

$$(S h_{(1)}\otimes S h_{(2)})(h_{(4)}\otimes h_{(3)}) (S h_{(5)}\otimes S h_{(6)}) = ((S h_{(1)})_{(2)}\otimes (S h_{(1)})_{(1)})(h_{(3)}\otimes h_{(2)})(S h_{(4)}\otimes S h_{(5)})$$

$$((S h_{(1)}\otimes S h_{(2)})\epsilon(h_{(3)}) = ((S h_{(1)})_{(2)}\otimes (S h_{(1)})_{(1)})(1\otimes\epsilon(h_{(2)}))$$

$$((S h_{(1)}\otimes S h_{(2)})\epsilon(h_{(3)}) = (\tau\circ\Delta)(S h_{(1)})(1\otimes\epsilon(h_{(2)})1) = (\tau\circ\Delta)(S h_{(1)}\epsilon(h_{(2)}))$$

$$(S h_{(1)}\otimes S h_{(2)})=(\tau\circ\Delta)(S h) = (S h)_{(2)}\otimes (S h)_{(1)}.$$

The axiom that must be satisfied by the antipode looks like a $k$-linear version of the identity satisfied by the inverse map in a group bimonoid: taking a group element $g$, duplicating by the diagonal map $\Delta$ to obtain $(g,g)$, 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 $Vect$, a Hopf algebra is a Hopf monoid in $Vect$.

Remark

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.

Properties

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 $V$ is the same as a Hopf monoid in the opposite category $V^{op}$ (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 $V$ is the same as a group object in $V$. 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 $V$ is the same as a group object in the opposite category $CMon(V)^{op}$, where $CMon(V)$ is the category of commutative monoids in $V$, hence a cogroup object in $CMon(V)$ (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 $Alg^{op}$ of affine schemes.

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

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

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

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

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

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

The theorem of Hopf modules

Hopf algebras can be characterized among bialgebras by the fundamental theorem of 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

• Milnor-Moore theorem

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 algebra	category of modules
$A$	$Mod_A$
$R$-algebra	$Mod_R$-2-module
sesquialgebra	2-ring = monoidal presentable category with colimit-preserving tensor product
bialgebra	strict 2-ring: monoidal category with fiber functor
Hopf algebra	rigid monoidal category with fiber functor
hopfish algebra (correct version)	rigid monoidal category (without fiber functor)
weak Hopf algebra	fusion category with generalized fiber functor
quasitriangular bialgebra	braided monoidal category with fiber functor
triangular bialgebra	symmetric monoidal category with fiber functor
quasitriangular Hopf algebra (quantum group)	rigid braided monoidal category with fiber functor
triangular Hopf algebra	rigid symmetric monoidal category with fiber functor
supercommutative Hopf algebra (supergroup)	rigid symmetric monoidal category with fiber functor and Schur smallness
form Drinfeld double	form Drinfeld center
trialgebra	Hopf monoidal category

2-Tannaka duality for module categories over monoidal categories

monoidal category	2-category of module categories
$A$	$Mod_A$
$R$-2-algebra	$Mod_R$-3-module
Hopf monoidal category	monoidal 2-category (with some duality and strictness structure)

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

monoidal 2-category	3-category of module 2-categories
$A$	$Mod_A$
$R$-3-algebra	$Mod_R$-4-module

As 3-vector spaces

A Hopf algebra structure on an associative algebra $A$ canonically defines on $A$ 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.

Relation to Frobenius algebras

Both Hopf algebras and Frobenius algebras are at the same time an algebra and a coalgebra, albeit with different additional structures and properties. Nevertheless, one may ask whether there is any relation between these two. This leads to the following result due to Larson & Sweedler (1969).

Proposition

Any finite-dimensional Hopf algebra can be endowed with the structure of a Frobenius algebra.

The finite-dimensional condition is expected since all Frobenius algebras are finite-dimensional. A nontrivial part of this result is that, while a finite-dimensional Hopf algebra $A$ already has a counit, this is not be the same counit that realizes $A$ as a Frobenius algebra. Rather, it is an integral (see Larson & Sweedler (1969) for details on this), which for finite-dimensional Hopf algebras always exist and are unique up to scaling.

In fact, more is true. In Fuchs, Schweigert & Stigner (2011) a symmetric special Frobenius algebra is constructed from a particular kind of Hopf algebra.

Proposition

Let $H$ be a finite-dimensional factorizable ribbon Hopf algebra, with multiplication $\mu$, counit $\epsilon$, comultiplication $\Delta$, and antipode $S$. Let $\Lambda$ be a left-integral element of $H$ and $\lambda$ a right-cointegral of $H$ (since $H$ is finite-dimensional, this is equivalent to an integral of the dual Hopf algebra $H^*$) such that $\lambda\circ\Lambda=1$. Then the dual vector space $H^*$ endowed with: a unit $\epsilon^*$, a multiplication $\Delta^*$, a counit $\Lambda^*$, and comultiplication

$$\Delta_F=((\text{id}_H\otimes (\lambda\circ \mu))\circ (\text{id}_H\otimes S\otimes\text{id}_H)\circ (\Delta\otimes\text{id}_H))^*$$

is a symmetric Frobenius object in the category $\text{Bimod}(H,H)$ of bimodules of $H$, and it is furthermore special iff $H$ is is semisimple.

Examples

• group algebra

• Steenrod algebra

• the ordinary homology of a finite type H-space (for instance a based loop space) is a Hopf algebra via its Pontrjagin ring-structure – see at homology of loop spaces for more

The Kac-Paljutkin $H_8$ Hopf algebra

The Kac-Paljutkin $H_8$ Hopf algebra was first described in Kac & Paljutkin (1966) and it is the Hopf algebra with the smallest dimension that is semisimple, noncommutative, and noncocommutative. It is also self-dual (see e.g. Burciu (2017)). One presentation is in terms of generators $\{x,y,z\}$ satisfying the relations $x^2=y^2=z^2=1$, $xz=zx$, $zy=yz$, $xyz=yx$, along with comultiplication

$$\Delta(x)=xe_0\otimes x+xe_1\otimes y$$

$$\Delta(y)=ye_1\otimes x+ye_0\otimes y$$

$$\Delta(z)=z\otimes z$$

(where $e_0=\frac{1}{2}(1+z)$, $e_1=\frac{1}{2}(1-z)$) whose counit is $\epsilon(x)=\epsilon(y)=\epsilon(z)=1$. The antipode map is

$$S(x)=xe_0+ye_1$$

$$S(y)=xe_1+ye_0$$

$$S(z)=z$$

. Its category of representations is the unique $\mathbb{Z}_2\times\mathbb{Z}_2$ Tambara-Yamagami category that admits a fiber functor and is not the representation category of some group.

Related concepts

• quasi-Hopf algebra

• triangular Hopf algebra

• quasitriangular Hopf algebra/quantum group

• hopfish algebra

• Hopf algebroid

  • commutative Hopf algebroid

• Hopf C-star algebra

• Hopf monoid

• Hopf monoidal category

• Hopf monad

• change of rings theorem

• Hopf ring spectrum

• Pontrjagin product

Tannaka duality for categories of modules over monoids/associative algebras

monoid/associative algebra	category of modules
$A$	$Mod_A$
$R$-algebra	$Mod_R$-2-module
sesquialgebra	2-ring = monoidal presentable category with colimit-preserving tensor product
bialgebra	strict 2-ring: monoidal category with fiber functor
Hopf algebra	rigid monoidal category with fiber functor
hopfish algebra (correct version)	rigid monoidal category (without fiber functor)
weak Hopf algebra	fusion category with generalized fiber functor
quasitriangular bialgebra	braided monoidal category with fiber functor
triangular bialgebra	symmetric monoidal category with fiber functor
quasitriangular Hopf algebra (quantum group)	rigid braided monoidal category with fiber functor
triangular Hopf algebra	rigid symmetric monoidal category with fiber functor
supercommutative Hopf algebra (supergroup)	rigid symmetric monoidal category with fiber functor and Schur smallness
form Drinfeld double	form Drinfeld center
trialgebra	Hopf monoidal category

2-Tannaka duality for module categories over monoidal categories

monoidal category	2-category of module categories
$A$	$Mod_A$
$R$-2-algebra	$Mod_R$-3-module
Hopf monoidal category	monoidal 2-category (with some duality and strictness structure)

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

monoidal 2-category	3-category of module 2-categories
$A$	$Mod_A$
$R$-3-algebra	$Mod_R$-4-module

References

• Kenneth Brown, Hopf algebras, lectures, pdf

• Nicolas Andruskiewitsch, Walter Ferrer Santos, The beginnings of the theory of Hopf algebras, Acta Appl Math 108 (2009) 3-17 [arXiv:0901.2460]

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 actions 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:

• Douglas Ravenel, W. Stephen Wilson, The Hopf ring for complex cobordism, Bull. Amer. Math. Soc. 80 (6) 1185 - 1189, November 1974 (doi:10.1016/0022-4049(77)90070-6, euclid, pdf)

  (for MU)

• Douglas Ravenel, appendix A1 of: Complex cobordism and stable homotopy groups of spheres, 1986 (pdf)

• Neil Strickland, Bott periodicity and Hopf rings, 1992 (pdf, pdf)

  (for topological K-theory)

• W. Stephen Wilson, Hopf rings in algebraic topology, Expositiones Mathematicae, 18:369–388, 2000 (pdf)

• Christoph Schweigert, Hopf algebras, quantum groups and topological field theory (2022) [pdf]

For Hopf algebras in generative linguistics, see:

• Matilde Marcolli, Noam Chomsky, Robert Berwick, Mathematical Structure of Syntactic Merge (arXiv:2305.18278)

• Matilde Marcolli, Robert Berwick, Noam Chomsky, Old and New Minimalism: a Hopf algebra comparison (arXiv:2306.10270)

The construction of a Frobenius algebra structure on finite-dimensional Hopf algebras due to

• Richard Larson, Moss Sweedler. An Associative Orthogonal Bilinear Form for Hopf Algebras. American Journal of Mathematics, Vol. 91, No. 1 (Jan., 1969), pp. 75-94 (20 pages). (doi)

See also

• Alfons Van Daele. Reflections on the Larson-Sweedler theorem for (weak) multiplier Hopf algebras (2024). (arXiv:2404.15046).

The proof that $H^*$ can be endowed with the structure of a symmetric special Frobenius object in $\text{Bimod}(H,H)$ is in

• Jürgen Fuchs, Christoph Schweigert, Carl Stigner?. Modular invariant Frobenius algebras from ribbon Hopf algebra automorphisms. Journal of Algebra 363, 1 August 2012, Pages 29-72. (doi)

On the $H_8$ Hopf algebra

• G. I. Kac, V. G. Paljutkin, Finite ring groups_, Trans. Moscow Math. Soc. 15 (1966), 251–294.

• Sebastian Burciu. Representations and conjugacy classes of semisimple quasitriangular Hopf algebras. (2017) ((arXiv:1709.02176).