Abstract
We give a geometric realization of the module category of a hereditary algebra of type \(\tilde {A}\). We work with oriented arcs to define a translation quiver isomorphic to the Auslander-Reiten quiver of the module category of type \(\tilde {A}\). To get a description of the module category, we introduce long moves between arcs. These allow us to include the infinite radical in the geometric description. Finally, our results can also be used to describe the corresponding cluster categories by taking unoriented arcs instead.
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Acknowledgements
KB thanks NTNU for supporting this project and the Mittag Leffler institute for inviting her to the program on Representation Theory in 2015. The authors thank the referee for careful reading and for several helpful suggestions.
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Appendix: Relations appearing in Theorem 3.4
Appendix: Relations appearing in Theorem 3.4
We use the geometric interpretation of relations to visualize the relations in Theorem 3.4
Relations (a) and (b) of the theorem are the usual mesh relations, they have already been described above.
Relations (c1),(c2)
Relation (c1) has on one side h compositions of elementary moves between preprojective arcs, followed by a long move and then g elementary moves in the tube of rank g, we can write it as
dropping the subscripts on the α and π. The arcs corresponding to these vertices all have lifts starting at 0∂, the Xi end at ∂′ and the Yi correspond to peripheral arcs. More precisely, we can choose a lift for X0, \(X_{0}=[-ghm_{\partial },ghm_{\partial ^{\prime }}]\). Under the arrows α, the arcs get lengthened by elementary moves fixing the ending point − ghm∂, up to the lift \([-ghm_{\partial },(ghm+h)_{\partial ^{\prime }}]\)Xh.
To prove that in our geometric set-up the first relation of (c1) holds, we can use the commuting triangle on the left hand side of Section 7.3.2 to see that ι0(0) ∘ α is a long move μh− 1 from Xh− 1 to Yg. Using this relation again, we see that α ∘ μh− 1 is a long move μh− 2 from Xh− 2 to Yg, etc. This proves that the composition \(\iota _{0}(0){\circ \alpha _{P}^{g}}\) is equal to a long move μ0 : X0 → Yg. Now we use the commuting triangle on the right hand side of Section 7.3.2 h times to iteratedly replace the long move from X0 composed with π by another long move from X0 to Yi, i = g − 1,…, 0.
The other three relations in (c1) and (c2) work completely analogously, using Sections 7.3.2, 7.3.3 and 7.3.4 respectively.
The relations in (e) rely on relation (f), so we will first consider the latter.
Relation (f)
For this, we need more work. Consider a tube \({\mathcal T}_{g}\) of rank g. We want to describe the effect of a composition of 2g elementary moves from an indecomposable object X of g to itself.
Let [a∂,b∂] be the arc (viewed in \(\operatorname {\mathbb U}\)), b ≥ a + 2. There are (at most) two elementary moves on [a∂,b∂].
Going down (up) from a peripheral arc in \(\operatorname {\mathbb U}\) corresponds to moving the endpoint (resp. the starting point) of the corresponding arc one step to the left, thus making it shorter (longer). We write fd for the elementary move downwards, fd : [a∂,b∂]↦[a∂, (b − 1)∂] and fu for the one going up, fu : [a∂,b∂]↦[(a − 1)∂,b∂]. Note that for b = a + 2, the image under fd is a boundary segment and hence zero.
For g ≥ 1 we write f↓g for the composition of g elementary moves downwards and f↑g for g consecutive elementary moves upwards. We then abbreviate the composition of g downwards with g upwards elementary moves by f↓↑g:
We will use the notations for these compositions of elementary moves within the universal cover \(\operatorname {\mathbb U}\), and also in the annulus.
The effect of a composition of g downwards with g upwards moves on peripheral arcs at the lower boundary is the following:
For arcs of \({\mathcal T}_{h}\), we define f↑↓h analogously, the effect of h elementary moves up followed by h elementary moves down. Since for endpoints at ∂′, elementary moves increase an endpoint by + 1, we get
In Pg,h, this shift by g or by + h is not visible, the effect of the compositions f↓↑g and f↑↓h on arcs in Pg,h is to move both endpoints of a peripheral arc around the boundary once (or zero, if the arc is close to the mouth of the tube):
Note that it follows from the (geometric) mesh relations, that the effect of g downwards moves combined with g upwords moves in \({\mathcal T}_{g}\) (h upwards moves with h downwards moves in \({\mathcal T}_{h}\)) is independent of the order in which this moves are done.
Relation (f) involves the four “top” vertices in Qm and arrows corresponding to long moves between them. We will first describe these vertices giving one lift in \(\operatorname {\mathbb U}\) for each of them. First recall that (ghm, 0)P = τ−ghm(0, 0)P, (ghm, 0)I = τghm(0, 0)I and observe that (0,g)g has as lift [0∂, 2∂], (0, 0)h has as lift \([-2_{\partial ^{\prime }},0_{\partial ^{\prime }}]\) (cf. Figure ?? for m = 2 to determine the latter).
In terms of lifts in \(\operatorname {\mathbb U}\), the four arrows ι∗(0), κ∗(0) of Qm can be described as long moves moving vertically between the two boundary components composed with a sequence of elementary moves.
1.1 A.1 ι∞(0): (ghm, 0)P → (2hm(n + 1) + h, 0)h
This arrow is a long move from an arc ∂ → ∂′ to a peripheral arc at ∂′. In geometric terms it is a rotation around the common ending point on ∂′ of the involved arcs. In terms of lifts in \(\operatorname {\mathbb U}\) with a common ending point:
The effect of ι∞(0) is to first use a long move μ, changing the starting point from ghm∂ by sending it vertically across to \(hhm_{\partial ^{\prime }}\) and then to use 2 + hm(n + 1) elementary moves νr still fixing the ending point on ∂′ to send this starting point along ∂′ to the left by subtracting 2 + hm(n + 1).
1.2 A.2 ι0(0): (ghm, 0)P → (2gm(n + 1) + g,g)g
The arrow ι0(0) is a long move around a common starting point on ∂. In terms of lifts:
The effect of ι0(0) is to first send the ending point of the arc from \((h+ghm)_{\partial ^{\prime }}\) to (g(1 + gm))∂ (long move μ) and then to send this point along ∂ to the right by adding 2 + gm(n + 1) (composition of 2 + gm(n + 1) elementary moves νr around the same starting point).
1.3 A.3 κ∞(0): (2hm(n + 1) + h, 0)h → (ghm, 0)I
This arrow corresponds to a rotation around the common starting point on ∂′. In \(\operatorname {\mathbb U}\),
we can describe the effect of κ∞(0). First, the endpoint of the arc on ∂′ is sent vertically across to (g + g2m)∂ under the long move μ and then under 2 − g − g(m(n + 1)) elementary moves μr, the endpoint is sent to the right to obtain \([(-2-h(m(n + 1)+hm)_{\partial ^{\prime }}, 2-ghm_{\partial }]\).
1.4 A.4 κ0(0): (2gm(n + 1) + g,g)g → (ghm, 0)I
κ0(0) corresponds to a rotation around the common ending point on ∂. In \(\operatorname {\mathbb U}\)
We first have a long move μ fixing the endpoints, sending the starting point from − ghm on ∂ across to − h2m on ∂′. This is then composed with − 2 + h + hm(n + 1) elementary moves μr to send the new starting point to the right and get the desired result.
1.5 A.5 \(\rho _{\infty }^{h}\pi _{\infty }^{h}\) and f↑↓h
In terms of arcs, the effect of the path \(\rho _{\infty }^{h}\pi _{\infty }^{h}\) is f↑↓h and the effect of \({\pi _{0}^{g}}{\rho _{0}^{g}}\) is f↓↑g.
Path \(\kappa _{\infty }(0)(\rho _{\infty }^{h}\pi _{\infty }^{h})^{j}\iota _{\infty }(0)\)
We use the lifts from above (Section I) describing ι∞(0) and compose with the remaining paths:
Applying κ∞(0) (Section A.3), this has the image
Path \(\kappa _{0}(0)({\pi _{0}^{g}}{\rho _{0}^{g}} )^{2m(n + 1)+ 1-j}\iota _{0}(0)\)
The arrow ι0(0) is described in Section A.2, we compose with the remaining paths:
Finally, applying κ0(0) (Section A.4), we get
With the projection to Pg,h in mind, we translate the resulting arc in \(\operatorname {\mathbb U}\) so that the images (11) and (10) of the two paths have the same endpoint. To the endpoint y∂ of (11) we add 2g(j − hm) (on boundary ∂). On ∂′, the corresponding translation is by + 2h(j − hm):
Hence the images of these arcs in Pg,h under the projection map are the same and relation (f) is satisfied.
Relation (e)
We only show the first claim κ0(0)ι0(0)(ghm,αn)P = 0, the second claim is completely analogous.
By (f), with j = 2m(n + 1) + 1, we know that \(\kappa _{0}(0)\iota _{0}(0) =\kappa _{\infty }{\rho _{\infty }^{h}\pi _{\infty }^{h}}^{h}\). In terms of arcs in the annulus, \(\rho _{\infty }^{h}\pi _{\infty }^{h}\) is the map f↑↓h from above, it sends any arc \(\pi [i_{\partial ^{\prime }},j_{\partial ^{\prime }}]\) with j − i ≥ h + 2 to itself. Relation (e) concerns a path going through (2m(n + 1)h, 0)h, the vertex corresponding to \(\alpha :=\pi [-2_{\partial ^{\prime }},(2hm(n + 1)-h)_{\partial ^{\prime }}]\) and this is high up in the tube \({\overline {\Gamma }}^{\infty }\), hence the effect of f↑↓h is to send this arc to itself. Applying f↑↓h to the arc 2m(n + 1) − 1 times still sends α to itself, however, the composition of all these elementary moves touches the mouth of the tube exactly once. In (e), this is precomposed with the elementary move corresponding to (ghm,αn)P. But this means that we can use Case B) from Section ?? to replace the elementary move in \({\overline {\Gamma }}^{P}\) (and the long move from \({\overline {\Gamma }}^{P}\)) by a long move to the arc \(\pi [-2_{\partial ^{\prime }}, (2hm(n + 1)+h-1)_{\partial ^{\prime }}]\) followed by an elementary move to α within the tube. Hence the path from α to itself, going all the way down to the mouth, is precomposed with one downwards move (in \({\overline {\Gamma }}^{\infty }\)). Then we use the mesh relations within \({\overline {\Gamma }}^{\infty }\) to push this all the way to the mouth, hitting the tube a second time just to the left of the other vertex at the mouth. That means that in our path, there are two shortest peripheral arcs. Between them, there is only a boundary segment (hence a zero object) and an arc of the form \([i_{\partial ^{\prime }},(i + 3)_{\partial ^{\prime }}]\). The zero relation at the mouth proves the claim (Section ??).
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Baur, K., Torkildsen, H.A. A Geometric Interpretation of Categories of Type \(\tilde {A}\) and of Morphisms in the Infinite Radical. Algebr Represent Theor 23, 657–692 (2020). https://doi.org/10.1007/s10468-019-09863-x
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DOI: https://doi.org/10.1007/s10468-019-09863-x