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We show that the irreducible components of any moduli space of semistable representations of a special biserial algebra are always isomorphic to products of projective spaces of various dimensions. This is done by showing that irreducible components of varieties of representations of special biserial algebras are isomorphic to irreducible components of products of varieties of circular complexes, and therefore normal, allowing us to apply recent results of the second and third authors on moduli spaces.
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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 $mgeq 3$ and $Ngeq 1$ are integers, and $Lambda_{m,N}$ is a self-injective special biserial $mathbf{k}$-algebra.
In this paper, we first show that for an acyclic gentle algebra A, the irreducible components of any moduli space of A-modules are products of projective spaces. Next, we show that the nice geometry of the moduli spaces of modules of an algebra does not imply the tameness of the representation type of the algebra in question. Finally, we place these results in the general context of moduli spaces of modules of Schur-tame algebras. More specifically, we show that for an arbitrary Schur-tame algebra A and theta-stable irreducible component C of a module variety of A-modules, the moduli space of theta-semi-stable points of C is either a point or a rational projective curve.
Let $k$ be an algebraically closed field, let $A$ be a finite dimensional $k$-algebra and let $V$ be a $A$-module with stable endomorphism ring isomorphic to $k$. If $A$ is self-injective then $V$ has a universal deformation ring $R(A,V)$, which is a complete local commutative Noetherian $k$-algebra with residue field $k$. Moreover, if $Lambda$ is also a Frobenius $k$-algebra then $R(A,V)$ is stable under syzygies. We use these facts to determine the universal deformation rings of string $Ar$-modules whose stable endomorphism ring isomorphic to $k$, where $Ar$ is a symmetric special biserial $k$-algebra that has quiver with relations depending on the four parameters $ bar{r}=(r_0,r_1,r_2,k)$ with $r_0,r_1,r_2geq 2$ and $kgeq 1$.
We determine the Gerstenhaber structure on the Hochschild cohomology ring of a class of self-injective special biserial algebras. Each of these algebras is presented as a quotient of the path algebra of a certain quiver. In degree one, we show that the cohomology is isomorphic, as a Lie algebra, to a direct sum of copies of a subquotient of the Virasoro algebra. These copies share Virasoro degree 0 and commute otherwise. Finally, we describe the cohomology in degree $n$ as a module over this Lie algebra by providing its decomposition as a direct sum of indecomposable modules.
The special linear representation of a compact Lie group G is a kind of linear representation of compact Lie group G with special properties. It is possible to define the integral of linear representation and extend this concept to special linear representation for next using.
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