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Hopf algebra

Idea

A Hopf algebra is a generalization of:

• the group algebra of a group

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

• the universal enveloping algebra of a Lie algebra

All these algebras are actually bialgebras, but furthermore they have a ‘antipode’ operation which reflects or mimics the inverse operation of a group.

Definition

A $k$-bialgebra $(A,m,\eta ,\Delta ,ϵ)$ with multiplication $m$, comultiplication $\Delta$, unit $\eta :k\to A$ and counit $ϵ:A\to k$ is a Hopf algebra if there exists a $k$-linear map

$$S:A\to A$$S : A \to A

called the antipode or coinverse such that $m\circ (\mathrm{id}\otimes S)\circ \Delta =m\circ (S\otimes \mathrm{id})\circ \Delta =\eta \circ ϵ$ (as a map $A\to A$). If an antipode exists then it is unique, just the way that if inverses exist in a monoid they are unique. The unit is group like, hence $S(1)1=1$, therefore $S(1)=1$. By linearity of $S$ this implies that $S\circ \eta \circ ϵ=\eta \circ ϵ$.

Proposition

The antipode is an antihomomorphism both of algebras and coalgebras (i.e. a homomorphism $S:A\to A_{\mathrm{op}}^{\mathrm{cop}}$).

Proof (algebra part)

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

$$S((hg{)}_{(1)})(hg{)}_{(2)}=ϵ(hg)$$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)

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

For the coalgebra part, notice first that $ϵ(h)1\otimes 1=\tau \circ \Delta (ϵ(h)1)=\tau \circ \Delta (S{h}_{(1)}{h}_{(2)})$. 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_{(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)})(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)})

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

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

$$(S{h}_{(1)}\otimes S{h}_{(2)})=(\tau \circ \Delta )(Sh)=(Sh{)}_{(2)}\otimes (Sh{)}_{(1)}.$$(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 $k$-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 $\mathrm{Vect}$, a Hopf algebra is a Hopf monoid in $\mathrm{Vect}$.

Hopf algebras versus 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 $V^{\mathrm{op}}$ (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 $\mathrm{CMon}(V{)}^{\mathrm{op}}$, where $\mathrm{CMon}(V)$ 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 ${\mathrm{Alg}}^{\mathrm{op}}$ 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 $k[-]:\mathrm{Set}\to k\mathrm{Mod}$ to the Hopf monoid $G$ in $\mathrm{Set}$.

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

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

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 $\mathrm{Set}\mid G\mid$.

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

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 $M(g_0,g_1)=g_0{g}_1$, $I=1$, $D(g)=g\otimes 1+1\otimes g$, $E(g)=0$, and the nifty Hopf antipodal operator $S(g)=-g$. Notice that the coalgebra operation $D,E$ depend only on $K\mathrm{Vect}\mid 𝔤\mid$.

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

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.

References

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, at www.ihes.fr

V. G. Drinfel’d, Quantum groups. Proceedings of the International Congress of Mathematicians 1986, Vol. 1, 2 798–820, AMS 1987.

G. Hochschild, Introduction to algebraic group schemes, 1971

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

John Milnor, J. 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-Verlag, New York-Berlin, 1979. xi+164 pp.

A Word of caution

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.