Jucys–Murphy elements and Grothendieck groups for generalized rook monoids

We consider a tower of generalized rook monoid algebras over the field $\mathbb{C}$ complex numbers and observe that Bratteli diagram associated to this is simple graph. construct modules describe Jucys-Murphy elements for algebras. Over an algebraically closed $\Bbbk$ positive characteristic $p$, utilizing algebras, $0\leq i\leq p-1$ we define corresponding $i$-restriction $i$-induction functors along with two extra functors. On direct sum $\mathcal{G}_{\mathbb{C}}$ Grothendieck groups module categories $\Bbbk$, these induce action tensor product universal enveloping algebra $U(\hat{\mathfrak{sl}}_p(\mathbb{C}))$ $\mathbb{C}[\mathcal{B}]$ bicyclic $\mathcal{B}$. Furthermore, prove isomorphic basic representation $U(\hat{\mathfrak{sl}}_{p}(\mathbb{C}))$ unique infinite-dimensional $\mathbb{C}[\mathcal{B}]$, also exhibit bialgebra. Under some natural restrictions on outline result monoids.