کد مقاله | کد نشریه | سال انتشار | مقاله انگلیسی | نسخه تمام متن |
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4583681 | 1630450 | 2016 | 28 صفحه PDF | دانلود رایگان |
For any positive integer d, we determine the irreducible components of the varieties that parametrize the d -dimensional representations of a local truncated path algebra Λ. Here Λ is a quotient KQ/〈the paths of lengthL+1〉 of a path algebra KQ, where K is an algebraically closed field, L is a positive integer, and Q is the quiver with a single vertex and a finite number r of loops. The components are determined in both the classical and the Grassmannian settings, Repd(Λ)Repd(Λ) and GRASSd(Λ)GRASSd(Λ). Our method is to corner the components by way of a twin pair of upper semicontinuous maps from Repd(Λ)Repd(Λ) to a poset consisting of sequences of semisimple modules.An excerpt of the main result is as follows. Given a sequence S=(S0,…,SL)S=(S0,…,SL) of semisimple modules with dim⨁0≤l≤LSl=ddim⨁0≤l≤LSl=d, let RepS be the subvariety of Repd(Λ)Repd(Λ) consisting of the points that parametrize the modules with radical layering SS. (The radical layering of a Λ-module M is the sequence (JlM/Jl+1M)0≤l≤L(JlM/Jl+1M)0≤l≤L, where J is the Jacobson radical of Λ.) Suppose the quiver Q has r≥2r≥2 loops. If d≤L+1d≤L+1, the variety Repd(Λ)Repd(Λ) is irreducible and, generically, its modules are uniserial. If, on the other hand, d>L+1d>L+1, then the irreducible components of Repd(Λ)Repd(Λ) are the closures of the subvarieties RepS for those sequences SS which satisfy the inequalities dimSl≤r⋅dimSl+1dimSl≤r⋅dimSl+1 and dimSl+1≤r⋅dimSldimSl+1≤r⋅dimSl for 0≤l
Journal: Journal of Algebra - Volume 464, 15 October 2016, Pages 198–225