Universal deformation rings for a class of self-injective special biserial algebras

Let $\mathbf{k}$ be an algebraically closed field of arbitrary characteristic, let $\Lambda$ be a finite dimensional $\mathbf{k}$-algebra and let $V$ be a $\Lambda$-module with stable endomorphism ring isomorphic to $\mathbf{k}$. If $\Lambda$ is self-injective, then $V$ has a universal deformation ring $R(\Lambda,V)$, which is a complete local commutative Noetherian $\mathbf{k}$-algebra with residue field $\mathbf{k}$. Moreover, if $\Lambda$ is further a Frobenius $\mathbf{k}$-algebra, then $R(\Lambda,V)$ is stable under syzygies. We use these facts to determine the universal deformation rings of string $\Lambda_{m,N}$-modules whose corresponding stable endomorphism ring is isomorphic to $\mathbf{k}$, and which lie either in a connected component of the stable Auslander-Reiten quiver of $\Lambda_{m,N}$ containing a module with endomorphism ring isomorphic to $\mathbf{k}$ or in a periodic component containing only string $\Lambda_{m,N}$-modules, where $m\geq 3$ and $N\geq 1$ are integers, and $\Lambda_{m,N}$ is a self-injective special biserial $\mathbf{k}$-algebra.