Exercise 14, § 7

Let K be a commutative field. Let M be the free monoid generated by two elements $x$ and $y$, and let A be the algebra of the monoid M over K. Let V be a vector space over K admitting a basis $(e_n)_{n\geqslant 1}$.

a) We define the structure of an A-module on V by setting $xe_1= 0,xe_p=e_{p-1}$ for $p >1$, and $ye_p=e_{2^p}$ for $p\geqslant 1$. Prove that V is a simple A-module.

b) Prove that the A-module V is faithful. (Let $z\in M$. For $n\in \mathbf{N}$, denote the integer $s$ such that $e_s=ze_n$ by $P_z(n)$. Observe that we have $P_{zt}= P_z\circ P_t$ for $z, t$ in M. Deduce by induction on the length of $z$ and $t$ that if $z\not=t$, the set of integers $n$ such that $P_z(n) = P_t(n)$ is finite.) c) Likewise, for every integer $r >2$, prove that by setting $xe_1= 0,xe_p=e_{p-1}$ for $p >1$, and $ye_p=e_{r^p}$ for $p\geqslant 1$, we define the structure of a simple and faithful A-module on V. Prove that these structures are pairwise nonisomorphic.