## Algebraic combinatorics Lecture 12 part 2: combinatorial Hopf algebra, examples

It is an attempt to algebraicize combinatorics. First we look at an example from Rota. Given a field ${k}$ with char ${\neq 2}$. Let ${\mathcal{R}}$ be the vector space over ${k}$ with a basis indexed by the set of all ranked posets.

Definition 2 A ranked poset is a finite poset with a unique smallest and largest element such that all maximal chains have the same length.

Examples are chains, hypercubes under the Hamming rank. I also thought of partitions with rank given by Cayley distance from the identity partition ${1^n}$.
${\mathcal{R}}$ has a natural grading given by the rank of the ranked posets. One can take product ${P.Q}$ of two ranked posets, with lexicographical grading presumably, which is an associative operation. ${\mathcal{R}}$ under this product is a graded algebra. Next we need a coproduct ${\Delta}$: the relation with categorical coproduct in terms of commutative diagram is yet unclear to me. Essentially ${\Delta}$ maps ${\mathcal{R} \rightarrow \mathcal{R} \otimes_k \mathcal{R}}$, whereas a product would reverse the arrow above. It needs to satisfy ${\Delta \otimes 1 \circ \Delta = 1 \otimes \Delta \circ \Delta}$ (coassociativity), which is similar to the associativity axiom for product in a commutative algebra. That’s why Hopf algebra is a coalgebra (or cogebra).
On ${\mathcal{R}}$, we define ${\Delta}$ on basis element ${P}$ a ranked poset:

$\displaystyle \Delta P = \sum_{z \in P} [0,z] \otimes [z,1] \ \ \ \ \ (7)$

where ${[0,z] = \{w \in P: 0 \le w \le z\}}$ is the subranked poset consisting of all elements between the smallest ${0}$ and ${z}$ in ${P}$. Note that ${\otimes}$ is not the same as the product of two ranked poset defined above. One can check that ${\Delta(P.Q) = \Delta(P).\Delta(Q)}$.
Also one defines a counit ${\epsilon: \mathcal{R} \rightarrow \mathcal{R}}$, which for the Rota example is given by

$\displaystyle \epsilon(P) = 1_{0_p = 1_p} \ \ \ \ \ (8)$

and zero otherwise.
Combinatorial Hopf algebras (CHA) have a character ${\zeta}$ in addition. In the ranked poset example, we take

$\displaystyle \zeta(P) = 1, \ \ \ \ \ (9)$

for all ${P}$. Then trivially ${\zeta(P) \zeta(Q) = \zeta(P.Q)}$. Honestly I haven’t seen any motivation for this.
Other examples of CHA:
${\mathcal{P}}$ a vector space with basis indexed by all finite posets, graded by ${|P|}$. ${P.Q = P \coprod Q}$ gives a graded algebra structure. Similar to the previous example,

$\displaystyle \Delta(P) = \sum_{I \subset P} I \otimes (P \setminus I) \ \ \ \ \ (10)$

where the sum is over all order ideals in P( definition: ${I \subset P}$ implies ${y \in I, x \le y \rightarrow x \in I}$. ), with ${\zeta(P) =1}$ again trivially.
We have a morphism of Hopf algebra ${J: \mathcal{P} \rightarrow \mathcal{R}}$ given on basis elements by ${J(P) = }$ the set of order ideals in P. The latter set has an obvious ranked poset structure. For more on Hopf algebra, S. Montgomery’s reference ‘Hopf algebra and their actions on rings’ is recommended (evil smile:). So now more formally,

Definition 3 A graded conected Hopf algebra over ${k}$ is a graded vector space ${A = A_0 \oplus A_1 \oplus \ldots}$ with an associative product operation that preserves grading, and also equipped with the following data

1. a coassociative(explained above) coproduct ${\Delta: A \rightarrow A \otimes A}$.
2. a counit ${\epsilon: A \rightarrow k}$, such that ${(\epsilon \otimes id) \circ \Delta = (id \otimes \epsilon) \circ \Delta = id}$. With these two data we have a coalgebra.
3. Also need ${\Delta(ab) = \Delta(a)\Delta(b)}$, where induced product on tensor space ${A \otimes A}$ is understood.

One nice thing about Hopf algebra is that if A is Hopf, and B any algebra, then ${Hom_{v.s.}(A,B)}$ has a product ${f.g}$ given by

$\displaystyle A \overset{\Delta}{\rightarrow} A \otimes A \overset{f \otimes g}{\rightarrow} B \otimes B \overset{m}{\rightarrow} B \ \ \ \ \ (11)$

where ${m}$ is the obvious tensor contraction product.
A character of Hopf algebra is an algebra map, ${\zeta : A \rightarrow k}$, with ${\zeta(ab) = \zeta(a) \zeta(b)}$, ${\zeta(1) =1}$. Having it gives a combinatorial Hopf algebra as mentioned earlier.