Let $P$ be a finite poset that we assume for simplicity to be bounded (that is it has a global maximum M and minimum m). Let k be a field, then the classical incidence algebra $kP$ has $k$-vector space basis elements $e_i$ for all $i \in P$ (the primitive idempotents) and $p_{l_2}^{l_1}$ for elements $l_1 < l_2$ in $P$.
Now we define a new finite dimensional algebra called the "Frobenius incidence algebra" T(P) associated to $P$ as follows: T(P) has vector space basis elements $e_i$ for all $i \in P$ (the primitive idempotents) and $p_{l_2}^{l_1}$ for elements $l_1 < l_2$ in $P$ and additionally the elements $p_{l_2}^{l_1}$ for $l_1 \geq l_2$. Thus the vector space dimension of $T(P)$ is two times the vector space dimension of $kP$. The relations the basis elements satsify look complicated but are quite natural (as in the incidence algebra case):
- $e_{l_1} p_{l_3}^{l_2}= \delta_{l_1,l_2} p_{l_3}^{l_2}$.
- $p_{l_2}^{l_1} e_{l_3} = \delta_{l_2 , l_3} p_{l_2}^{l_1}$.
- $e_{l_1} e_{l_2} = \delta_{l_1,l_2} e_{l_1}$.
- $p_{l_2}^{l_1} p_{l_4}^{l_3} = \delta_{l_2, l_3} p_{l_4}^{l_1}$.
- $p_{l}^{l} p_{l_2}^{l_1}=0$.
- $p_{l_2}^{l_1} p_{l}^{l}=0$.
- $ p_M^a p_m^M p_b^m$=0 for all $a \nleq b$ and $b \nleq a$.
In fact $T(P)$ is isomorphic to the trivial extension algebra of the incidence algebra $kP$ and thus $T(P)$ is a (symmetric) Frobenius algebra.
Now $T(P)$ has additionally to the $k$-algebra structure also a coalgebra structure $(\lambda, \delta)$ as follows:
$\lambda$ is defined by $\lambda(p_{l_2}^{l_1})=1$ if and only if $l_1=l_2$ and zero else, and $\lambda(e_i)=0$.
The coproduct $\delta$ is given by $\delta( p_{l_2}^{l_1})= \sum\limits_{t}^{}{p_t^{l_1} \otimes p_{l_2}^t}$, where the sum is over all $t$ that are comparable to $l_1$ and $l_2$, and $\delta(e_q)= p_q^q \otimes p_q^q + e_q \otimes e_q$.
Sadly $\lambda$ is never an algebra map when $P$ has at least two elements and thus this does not give rise to a bialgebra structure. The next questions are for general $P$ but might be more interesting for special choices such as $P$ being the Boolean poset.
Question 1: Is there more algebraic structure on $T(P)$? Maybe there is another coalgebra structure that gives rise to a bialgebra or even (graded) Hopfalgebra strucutre?
For example when $P$ is the chain with two elements, $T(P)$ in characteristic 3 is isomorphic to the symmetric group algebra $k S_3$ and thus has a Hopf algebra structure.
Question 2: Are the interesting combinatorial applications of the algebra $T(P)$ and the coalgebra structure on $T(P)$?
Note that the Cartan matrix of $T(P)$ is given by $C+C^{T}$ when $C$ is the lequal matrix of $P$. For example for the Boolean poset an interesting connection to the Catalan numbers is suggested here: Factorisation of a polynomial from the Boolean algebra
