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This shouldn't be surprising, though: path algebras of quivers have global dimension one, so you shouldn't expect their derived categories to agree with derived categories of sheaves on higher-dimensional …

Edited to add: oh, another thing: for Dynkin quivers, there is a way to construct any indecomposable by applying a sequence of "reflection functors" starting from a simple indecomposable, but the sequence …

This sounds wrong to me. I think $D^b(Q)$ should be replaced by the derived category of finite length modules over the corresponding preprojective algebra of affine type. Homological mirror symmetry …

For me, path algebras of quivers have provided a lovely setting in which to learn about homological algebra, because the objects involved are simple enough that you can understand them quite concretely …

Edited to be more of an answer. But, unfortunately, to be quite wrong. I apologize for having been borne away by my enthusiasm. The picture of the free modular lattice above shows thirty elements. …

Let $Q$ be a quiver, and let $d=(d_i)$ be a dimension vector. We can consider Rep($Q,d$), the affine space consisting of representations of $Q$ with dimension vector $d$. The general linear $GL(d)= \p …

There is another way to relate representations of $Q$ to representations of $Q'$: reflection functors. These are quite easy to describe combinatorially. One downside is that the way they work is by …

The quiver given in the question has five simple modules, six which correspond to a single arrow, and the remaining representations have support as follows: 123, 124, 125, 235, 345, 1235, 12235 (note …

For other tame quivers with no relations over an algebraically closed ground field, the situation is slightly worse: the natural indexing set for the representations whose dimension vector is the null …

Let $Q$ be a quiver without loops (cycles of length 1). Kac proved that if $K$ is algebraically closed, the dimension vectors of indecomposable representations of $Q$ over $K$ are exactly the positiv …

Dag has already answered the case where the quiver is finite and acyclic, and given a conjecture in the case that cycles are allowed. I will prove his conjecture. Suppose we have an element $x$ of …

The AR quiver of the regular representations of an affine quiver consists of infinitely many "tubes". A tube of rank $r$ has $r$ modules on what you call the border. Let me number them $B_1, B_2, \dot …

As a module over $kQ$, a finite-type preprojective algebra is a direct sum of each of the indecomposable $kQ$-modules once. Thus, the total dimension is the sum over all positive roots of the height …

There is an example in section 3.4 of Riedtmann's paper "Degenerations for representations of quivers with relations", Ann. sci. Éc. Norm. Sup. v. 19 (1986), 275-301. …

This answer perhaps says things that are all obvious to the OP. $\textrm{Ext}^i(A,B)$ is a vector space over the ground field, so its cardinality is $q^d$ where $d$ is the dimension of the vector sp …