Definition of $C_L$ for people who like number theory:

Let $m$ be a number with prime factorisation $m=p_1^{n_1} ... p_r^{n_r}$ with $n_i>0$. Define $I_m$ to be the incidence algebra of the divisor lattice of $m$. Up to isomorphism this just depence on the $n_i$ and not on the primes so lets define with $L=[n_1+1,...,n_r+1]$, $I_L$ as the incidence algebra of the divisor lattice of a number $m$ with prime factorisation $m=p_1^{n_1} ... p_r^{n_r}$. Define $C_L$ as the trivial extension algebra of $I_L$. For the definition of trivial extension algebra see for example https://math.stackexchange.com/questions/229412/trivial-extension-of-an-algebra (together with a proof that this is always a Frobenius algebras) or the textbook on Frobenius algebras I by Skowronski and Yamagata.

Definition of $C_L$ for people who like Nakayama algebras:

For a natural number $n \geq 2$ denote by $A_n$ the hereditary Nakayama algebra with dimension $n$ given by quiver and relations over a field $K$ (that is the Nakayama algebra with Kupisch series $[n,n-1,...,2,1]$). For a list of integers $L=[n_1,...,n_r]$. Define the algebra $C_L$ as the trivial extension of the algebra $A_{n_1} \otimes_K \cdots \otimes_K A_{n_r}$.

Questions:

Is $C_L$ isomorphic to a known/studied algebra from the literature? Experiments with small lists suggest that the algebra is always periodic or at least all simple modules are periodic with the same period. (Recall that a module $M$ is called periodic with period $n$ when $n$ is the smallest integer such that $\Omega^n(M) \cong M$. An algebra is called periodic in case the regular module is periodic as a bimodule.) The period seems to behave in a strange way. What could it be (in case such a finite periodic exists)?

Here some examples, where after the list $L$ the number is the period of a simple module of $C_L$. Sadly even with a computer it takes long to do such calculations.

$L=[t]: 2t$ for $t \geq 1$ (this is well known)

$L=[2,2]: 5$

$L=[3,2]: 22$

$L=[4,2]:29$

$L=[5,2]: 12$ (this was surprising)

$L=[6,2]: 43$

$L=[4,3]:42$

$L=[3,3]: 8$

$L=[2,2,2]: 6$

$L=[3,2,2]: 26$

Guesses what the period might be for general $L$ are also welcome. I should say that it is not too good tested yet as even for those small lists it takes several minutes to do the calculations with a computer.

More general, one might ask what those periods are (in case they exist) for the trivial extension algebra of tensor products of hereditary path algebras of Dynkin type. For example the trivial extension of $k D_4 \otimes_K k A_2$ has the property that all simple modules have period equal to 6.