symmetric monoidal (∞,1)-category of spectra
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.
A $k$-bialgebra $(A,m,\eta,\Delta,\epsilon)$ with multiplication $m$, comultiplication $\Delta$, unit $\eta: k\to A$ and counit $\epsilon:A\to k$ is called a Hopf algebra if there exists a $k$-linear function
called the antipode or coinverse such that
(as a map $A\to A$).
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 $S\circ\eta\circ\epsilon = \eta\circ\epsilon$.
The antipode is an antihomomorphism both of algebras and coalgebras (i.e. a homomorphism $S:A\to A^{cop}_{op}$).
In Sweedler notation, for any $g,h\in A$,
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
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 $Vect$.
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.
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 $V^{op}$ (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:
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.
A commutative Hopf monoid in a symmetric monoidal category $V$ is the same as a group object in $CMon(V)^{op}$, where $CMon(V)$ is the category of commutative monoids in $V$. 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):
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 $Set$.
The function algebra $k(G)$ (the set of functions $G\to k$), with comultiplication given by precomposition with the group operation
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.
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 $2$ very different ways. Both ways have a discrete version and a smooth version.
Given a (finite, discrete) group $G$ and a ground ring (field?) $K$, then the group ring $K[G]$ is a cocommutative Hopf algebra, with $M(g_0,g_1) = g_0 g_1$, $I = 1$, $D(g) = g \otimes g$, $E(g) = 1$, and the nifty Hopf antipodal operator $S(g) = g^{-1}$. Notice that the coalgebra operations $D,E$ depend only on $Set|G|$.
Given a (finite, discrete) group $G$ and a ground ring (field?) $K$, then the function ring $Fun(G,K)$ is a commutative Hopf algebra, with $M(f_0,f_1)(g) = f_0(g)f_1(g)$, $I(g) = 1$, $D(f)(g,h) = f(g h)$, $E(f) = f(1)$, and the nifty Hopf antipodal operator $S(f)(g) = f(g^{-1})$. Notice that the algebra operations $M,I$ depend only on $Set|G|$.
Given a (simply connected) Lie group $G$ and the complex (real?) field $K$, then the universal enveloping algebra $U(G)$ is a cocommutative Hopf algebra, with $M(\mathbf{g}_0,\mathbf{g}_1) = \mathbf{g}_0 \mathbf{g}_1$, $I = 1$, $D(\mathbf{g}) = \mathbf{g} \otimes 1 + 1 \otimes \mathbf{g}$, $E(\mathbf{g}) = 0$, and the nifty Hopf antipodal operator $S(\mathbf{g}) = -\mathbf{g}$. Notice that the coalgebra operation $D,E$ depend only on $K Vect|\mathfrak{g}|$.
Given a (compact) Lie group $G$ and the complex (real?) field $K$, then the algebraic function ring $Anal(G)$ is a cocommutative Hopf algebra, with $M(f_0,f_1)(g) = f_0(g) f_1(g)$, $I(g) = 1$, $D(f)(g,h) = f(g h)$, $E(f) = f(1)$, and the nifty Hopf antipodal operator $S(f)(g) = f(g^{-1})$. Notice that the algebra operations $M,I$ depend only on $Anal Man|G|$.
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.
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 |
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 |
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.
For a diagrammatic definition of a Hopf algebra, see the Wikipedia entry.
Eiichi Abe, Hopf algebras, Cambridge UP 1980.
Pierre Cartier, A primer on Hopf algebras, IHES 2006, 81p (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
S. Majid, Foundations of quantum group theory, Cambridge University Press 1995, 2000.
John Milnor, John Moore, The structure of Hopf algebras, Annals of Math. 81 (1965), 211-264.
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.
M. 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
and discussed in detail in
Discussion with an eye towards stable homotopy theory and the Steenrod algebra is in