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Register a new userOn a deformation theory of finite dimensional modules over repetitive algebras
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Let $Lambda$ be a basic finite dimensional algebra over an algebraically closed field $mathbf{k}$, and let $widehat{Lambda}$ be the repetitive algebra of $Lambda$. In this article, we prove that if $widehat{V}$ is a left $widehat{Lambda}$-module with finite dimension over $mathbf{k}$, then $widehat{V}$ has a well-defined versal deformation ring $R(widehat{Lambda},widehat{V})$, which is a local complete Noetherian commutative $mathbf{k}$-algebra whose residue field is also isomorphic to $mathbf{k}$. We also prove that $R(widehat{Lambda},widehat{V})$ is universal provided that $underline{mathrm{End}}_{widehat{Lambda}}(widehat{V})=mathbf{k}$ and that in this situation, $R(widehat{Lambda},widehat{V})$ is stable after taking syzygies. We apply the obtained results to finite dimensional modules over the repetitive algebra of the $2$-Kronecker algebra, which provides an alternative approach to the deformation theory of objects in the bounded derived category of coherent sheaves over $mathbb{P}^1_{mathbf{k}}$
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Let $mathbf{k}$ be a field of arbitrary characteristic, let $Lambda$ be a finite dimensional $mathbf{k}$-algebra, and let $V$ be a finitely generated $Lambda$-module. F. M. Bleher and the third author previously proved that $V$ has a well-defined versal deformation ring $R(Lambda,V)$. If the stable endomorphism ring of $V$ is isomorphic to $mathbf{k}$, they also proved under the additional assumption that $Lambda$ is self-injective that $R(Lambda,V)$ is universal. In this paper, we prove instead that if $Lambda$ is arbitrary but $V$ is Gorenstein-projective then $R(Lambda,V)$ is also universal when the stable endomorphism ring of $V$ is isomorphic to $mathbf{k}$. Moreover, we show that singular equivalences of Morita type (as introduced by X. W. Chen and L. G. Sun) preserve the isomorphism classes of versal deformation rings of finitely generated Gorenstein-projective modules over Gorenstein algebras. We also provide examples. In particular, if $Lambda$ is a monomial algebra in which there is no overlap (as introduced by X. W. Chen, D. Shen and G. Zhou) we prove that every finitely generated indecomposable Gorenstein-projective $Lambda$-module has a universal deformation ring that is isomorphic to either $mathbf{k}$ or to $mathbf{k}[![t]!]/(t^2)$.
Let $mathbf{k}$ be field of arbitrary characteristic and let $Lambda$ be a finite dimensional $mathbf{k}$-algebra. From results previously obtained by F.M Bleher and the author, it follows that if $V^bullet$ is an object of the bounded derived category $mathcal{D}^b(Lambdatextup{-mod})$ of $Lambda$, then $V^bullet$ has a well-defined versal deformation ring $R(Lambda, V^bullet)$, which is complete local commutative Noetherian $mathbf{k}$-algebra with residue field $mathbf{k}$, and which is universal provided that $textup{Hom}_{mathcal{D}^b(Lambdatextup{-mod})}(V^bullet, V^bullet)=mathbf{k}$. Let $mathcal{D}_textup{sg}(Lambdatextup{-mod})$ denote the singularity category of $Lambda$ and assume that $V^bullet$ is a bounded complex whose terms are all finitely generated Gorenstein projective left $Lambda$-modules. In this article we prove that if $textup{Hom}_{mathcal{D}_textup{sg}(Lambdatextup{-mod})}(V^bullet, V^bullet)=mathbf{k}$, then the versal deformation ring $R(Lambda, V^bullet)$ is universal. We also prove that certain singular equivalences of Morita type (as introduced by X. W. Chen and L. G. Sun) preserve the isomorphism class of versal deformation rings of bounded complexes whose terms are finitely generated Gorenstein projective $Lambda$-modules.
Let $mathbf{k}$ be an algebraically closed field, let $Lambda$ be a finite dimensional $mathbf{k}$-algebra, and let $widehat{Lambda}$ be the repetitive algebra of $Lambda$. For the stable category of finitely generated left $widehat{Lambda}$-modules $widehat{Lambda}$-underline{mod}, we show that the irreducible morphisms fall into three canonical forms: (i) all the component morphisms are split monomorphisms; (ii) all of them are split epimorphisms; (iii) there is exactly one irreducible component. We next use this fact in order to describe the shape of the Auslander-Reiten triangles in $widehat{Lambda}$-underline{mod}. We use the fact (and prove) that every Auslander-Reiten triangle in $widehat{Lambda}$-underline{mod} is induced from an Auslander-Reiten sequence of finitely generated left $widehat{Lambda}$-modules.
Khovanov-Lauda-Rouquier algebras of finite Lie type come with families of standard modules, which under the Khovanov-Lauda-Rouquier categorification correspond to PBW-bases of the positive part of the corresponding quantized enveloping algebra. We show that there are no non-zero homomorphisms between distinct standard modules and all non-zero endomorphisms of a standard module are injective. We obtain applications to extensions between standard modules and modular representation theory of KLR algebras.
The deformed current Lie algebra was introduced by the author to study the representation theory of cyclotomic q-Schur algebras at q=1. In this paper, we classify finite dimensional simple modules of deformed current Lie algebras.
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