Hopf algebras examples

Question

Following Richard Borcherds' questions 34110 and 61315, I'm looking for interesting examples of Hopf algebras for an introductory Hopf algebras graduate course.

Some of the examples I know are well-known:

• Examples related to groups and Lie algebras,
• Sweedler and Taft Hopf algebras,
• Drinfeld-Jimbo quantum groups.

(These examples can be found for example in Montgomery's book on Hopf algebras.) Other interesting examples are Lusztig small quantum groups and the generalized small quantum groups of Andruskiewitsch and Schneider.

Therefore I'm looking for interesting and non-standard examples maybe related to combinatorics or to other branches of mathematics.

Answer 1

Symmetric functions, quasisymmetric functions, Connes-Kreimer algebra are all examples of combinatorial Hopf algebras. There are many more.

The original example is the cohomology ring of a Lie group.

Also Steenrod algebra and the analogue for any cohomology theory.

Answer 2

If you're interested in Hopf algebras in categories other than $\mathrm{Vect}$, you can look at the exterior algebra as a Hopf algebra in $\mathrm{SVect}$, the category of super vector spaces with degree-preserving morphisms. More precisely, let $V$ be a purely odd vector space (i.e. $V_0 = 0$ and $V_1 = V$) and form the exterior algebra $\Lambda(V)$ with its natural $\mathbb{Z}/2$ grading. This is a superalgebra, i.e. an algebra in $\mathrm{SVect}$. Then $\Lambda(V) \underline{\otimes} \Lambda(V)$ is an algebra in $\mathrm{SVect}$, where $\underline{\otimes}$ is the graded tensor product of graded algebras.

Now consider the map $\Delta : V \to \Lambda(V) \underline{\otimes} \Lambda(V)$ given by $$\Delta(v) = v \otimes 1 + 1 \otimes v.$$ With the sign conventions coming from the graded tensor product, you get $\Delta(v)^2 = 0$, and so according to the universal property of the exterior algebra, $\Delta$ extends to an algebra homomorphism $\Delta : \Lambda(V) \to \Lambda(V) \underline{\otimes} \Lambda(V)$. Coassociativity is clear. You can get the counit and antipode similarly using the universal property.

---

Another good example is the shuffle Hopf algebra, which is discussed in this question. Let $V$ be a vector space and $T(V)$ its tensor algebra. The shuffle Hopf algebra is a Hopf structure on $T(V)$ which uses neither the standard algebra nor coalgebra structures on the tensor algebra.

The comultiplication is given by deconcatenation: $$ \Delta(v_1 \dots v_n) = \sum_{j=1}^{n+1} v_1 \dots v_{j-1} \otimes v_j \dots v_n, $$ while the multiplication is given by the shuffle product: $$ (v_1 \dots v_k) \cdot (v_{k+1} \dots v_n) = \sum_{\sigma \in S_{k,n-k}} v_{\sigma^{-1}(1)} \dots v_{\sigma^{-1}(n)},$$ where $S_{k,n-k}$ is the set of $(k,n-k)$ shuffle permutations, i.e. $$\sigma(1) < \dots < \sigma(k)$$ and $$\sigma(k+1) < \dots < \sigma(n).$$ I haven't really worked much with the shuffle algebra myself, but the answers to the question linked above have some discussion of what it is good for.

Answer 3

A couple people mentioned the Steenrod algebra briefly, but you can do a few more topologically-related things:

• The subalgebras $\mathcal{A}(n)$ of the Steenrod algebra generated by $Sq^1, ..., Sq^n$ are neat. In particular, it is a good exercise in cohomology to compute $Ext_{\mathcal{A}(n)}(k,k)$. (One can do a minimal resolution and try to look for a pattern, and then prove that it works using a spectral sequence.)

• You can show that the Hopf algebras given by $\mathbb{Z}[c_1, c_2, ...]$ ($c_i$ living in degree 2i) and $\mathbb{Z}/2 [w_1, w_2, ...]$ ($w_i$ living in degree i) and comultiplications given by $y_n \mapsto \sum y_i \otimes y_j$ on the generators, are self-dual Hopf algebras and explicitly describe the relationship between itself and the dual. This is neat in and of itself, but then you can mention that these results lead to quick calculations of $H_*MU$ and $H_*MO$ as comodules over the dual of the Steenrod algebra, and thus allow for computations of cobordism groups via the Adams spectral sequence. This self-duality can also be used for a quick proof of the Bott periodicity theorem, though the only reference I know of for this is not yet published (by May), though it will be soon.

• If you're doing cohomology, it's always nice to do the cohomology of an exterior algebra; i.e. a Hopf algebra that's an exterior algebra on primitive generators. It's a very easy result, but you can use it to compute other things via spectral sequences.

I know I've forgotten several things I wanted to mention... if I remember them, I'll edit them in.

Answer 4

I will elaborate on Bruce's examples and add a couple of my own.

Of course, the homology and cohomology of topological groups over a field are good examples.

For each prime prime $p$ the Steenrod algebra $\mathcal{A}_p$ which is the algebra of endomorphisms of the cohomology theory $H^*(-;\mathbb{F}_p)$. The cohomology of this Hopf algebra is the $E_2$ term of a spectral sequence, due to Adams, converging to the $p$-completed stable homotopy groups of spheres.

The functions on any affine algebraic groups over a field are another family of examples.

Formal group laws over a field $k$. You can read about these in Husemoller's book.

The rational homotopy groups of connected topological group or more generally an $H$-space is a Lie algebra. A nice result of Milnor-Moore shows that the universal enveloping algebra of this Lie algebra is isomorphic as Hopf algebras to the rational homology of the space.

Answer 5

There are lots of good combinatorial examples that are not hard to define: e.g., posets, matroids, quasisymmetric functions, graphs. A few places to look: W. Schmitt, Incidence Hopf Algebras, J. Pure Appl. Algebra 96 (1994), no. 3, 299–330. W. Schmitt, Hopf algebra methods in graph theory, J. Pure Appl. Algebra 101 (1995), no. 1, 77–90. M. Aguiar, N. Bergeron, and F. Sottile, Combinatorial Hopf algebras and generalized Dehn-Sommerville relations, Compos. Math. 142 (2006), no. 1, 1–30.

Answer 6

My favorite Hopf algebra is the following:

take $x$, $d$ two free variables, fix a field $k$ and consider $H$ the $k$-algebra generated by $x^{\pm 1}$, $d$ with relations

$xd+dx=0$,

$d^2=0$

This is a Hopf algebra if one defines the comultiplication $\Delta(x)=x\otimes x$,

$\Delta d=d\otimes 1+x\otimes d$

It is a kind of infinite dimensional version of the Taft algebra. The reason why it is my favorite is because the category of H-comodules is exactely d.g. $k$-vector spaces.

Notice that $H$ contains $k[x^{\pm 1}]=k[\mathbb Z]$ as sub (Hopf) algebra and the evaluation "$d=0$" gives a map $H\to k[\mathbb Z]$, in particular, every $H$-comodule is a $k[\mathbb Z]$-comodule = $\mathbb Z$-graded vector space.

Analyzing the role of $d$ in the general coaction of a comodule $M$ gives you the differential of $M$.

It is a very nice exercise to check whether the "choice" in the definition

"$\Delta d=d\otimes 1+x\otimes d$"

or

$\Delta d=d\otimes x+1\otimes d$

gives you that the differential raises or decrease the degree by one.

If $C$ is a d.g. coalgebra, then $C\#H$ is a usual (i.e. no grading, no differential considerations in the definition of coalgebra), and d.g. $C$- comodules are the same as (non graded, nor differential) $C\#H$-comodules.

Answer 7

The Hall algebra approach to quantum groups is important as well. To any finitary hereditary $\mathbb{F}_q$-linear abelian category $\mathcal{A}$ one can associate a so called Hall algebra $\mathbb{H}(\mathcal{A})$. The Hall algebra is a Hopf algebra for categories satisfying the finite subobject condition (more general definitions exist as well).

In particular, considering the category $\mathcal{A}$ of finite dimensional representations of the quiver $A_n$ yields $\mathbb{H}(\mathcal{A})\cong U_v(\mathfrak{sl}_{n+1})$ where $v=q^{\frac{1}{2}}$. It is absolutely remarkable that Hall algebra (which essentially describes the extension-structure of a category) recovers these quantum groups.

In principle you can generate many Hopf algebras by plugging in suitable categories, though I admit I only looked at familiar examples.

Answer 8

Hopf incidence algebras are not well understood and could be an important example as it deals with matrix multiplication. Fourier transform for these are not well understood.

Answer 9

There's a (unique) semisimple noncocommutative Hopf algebra of dimension 8 that makes a nice example. (Unfortunately I don't remember where to find information on it at the moment, I learned about it in a survey paper of Susan Montgomery's.)

Answer 10

To complement the answer about Hopf algebras in categories other than Vect: in the monoidal categories of stereotype spaces, (Ste,$\circledast$) and (Ste,$\odot$) (which generalize the category (Vect,$\otimes$) of vector spaces) there are lots of examples of Hopf algebras. Some of them are generalizations of the standard constructions, like stereotype group algebras, some others generalize quantum groups, some others appear as envelopes of already constructed Hopf algebras. A detailed explanation can be found here.