Aslak Bakke Buan
Current Organizations / Institutes
The organizations/institutes from the scholar's latest article
Research Subjects
Aslak Bakke Buan has a total of 18 co-authors. Here are the top co-authors by number of shared publications:
- Idun Reiten
Department of Mathematical Sciences, Norwegian University of Science and Technology, 7491 Trondheim, Norway
19 shared publications - Bethany Rose Marsh
School of Mathematics University of Leeds Leeds UK
12 shared publications - Hugh Thomas
University of New Brunswick
4 shared publications - Yu Zhou
Yau Mathematical Sciences Center, Tsinghua University, Beijing, China
4 shared publications - Gordana Todorov
Department of Mathematics, Northeastern University, Boston, USA
3 shared publications - Osamu Iyama
Graduate School of Mathematics, Nagoya University, Furocho, Chikusaku, Nagoya 464-8602, Japan
3 shared publications
Latest Publications
- Open Access
Decorated marked surfaces III: The derived category of a decorated marked surface
We study the Ginzburg dg algebra $\Gamma_\mathbf{T}$ associated to the quiver with potential arising from a triangulation $\mathbf{T}$ of a decorated marked surface $\mathbf{S}_\bigtriangleup$, in the sense of Qiu. We show that there is a canonical way to identify all finite dimensional derived categories $\mathcal{D}_{fd}(\Gamma_\mathbf{T})$, denoted by $\mathcal{D}_{fd}(\mathbf{S}_\bigtriangleup)$. As an application, we show that the spherical twist group $\operatorname{ST}(\mathbf{S}_\bigtriangleup)$ associated to $\mathcal{D}_{fd}(\mathbf{S}_\bigtriangleup)$ acts faithfully on its space of stability conditions.Open AccessDecorated marked surfaces III: The derived category of a decorated marked surface
We study the Ginzburg dg algebra $\Gamma_\mathbf{T}$ associated to the quiver with potential arising from a triangulation $\mathbf{T}$ of a decorated marked surface $\mathbf{S}_\bigtriangleup$, in the sense of Qiu. We show that there is a canonical way to identify all finite dimensional derived categories $\mathcal{D}_{fd}(\Gamma_\mathbf{T})$, denoted by $\mathcal{D}_{fd}(\mathbf{S}_\bigtriangleup)$. As an application, we show that the spherical twist group $\operatorname{ST}(\mathbf{S}_\bigtriangleup)$ associated to $\mathcal{D}_{fd}(\mathbf{S}_\bigtriangleup)$ acts faithfully on its space of stability conditions. - 14 June 2017
- Published by Springer Nature in Algebras and Representation Theory
- Vol. 21 (1) , 181-194
Endomorphism Algebras of 2-term Silting Complexes
- 14 June 2017
- Published by Springer Nature in Algebras and Representation Theory
- Vol. 21 (1) , 181-194
Endomorphism Algebras of 2-term Silting Complexes
- Open Access
Endomorphism algebras of 2-term silting complexes
We study possible values of the global dimension of endomorphism algebras of 2-term silting complexes. We show that for any algebra $A$ whose global dimension $\mathop{\rm gl. dim}\nolimits A\leq 2$ and any 2-term silting complex $\mathbf{P}$ in the bounded derived category ${D^b(A)}$ of $A$, the global dimension of $\mathop{\rm End}\nolimits_{D^b(A)}(\mathbf{P})$ is at most 7. We also show that for each $n>2$, there is an algebra $A$ with $\mathop{\rm gl. dim}\nolimits A=n$ such that ${D^b(A)}$ admits a 2-term silting complex $\mathbf{P}$ with $\mathop{\rm gl. dim}\nolimits \mathop{\rm End}\nolimits_{D^b(A)}(\mathbf{P})$ infinite.Open AccessEndomorphism algebras of 2-term silting complexes
We study possible values of the global dimension of endomorphism algebras of 2-term silting complexes. We show that for any algebra $A$ whose global dimension $\mathop{\rm gl. dim}\nolimits A\leq 2$ and any 2-term silting complex $\mathbf{P}$ in the bounded derived category ${D^b(A)}$ of $A$, the global dimension of $\mathop{\rm End}\nolimits_{D^b(A)}(\mathbf{P})$ is at most 7. We also show that for each $n>2$, there is an algebra $A$ with $\mathop{\rm gl. dim}\nolimits A=n$ such that ${D^b(A)}$ admits a 2-term silting complex $\mathbf{P}$ with $\mathop{\rm gl. dim}\nolimits \mathop{\rm End}\nolimits_{D^b(A)}(\mathbf{P})$ infinite. - Open Access
A silting theorem
We give a generalization of the classical tilting theorem. We show that for a 2-term silting complex $\mathbf{P}$ in the bounded homotopy category $K^b(\mathop{\rm proj}\nolimits A)$ of finitely generated projective modules of a finite dimensional algebra $A$, the algebra $B = \mathop{\rm End}\nolimits_{K^b(\mathop{\rm proj}\nolimits A)}(\mathbf{P})$ admits a 2-term silting complex $\mathbf{Q}$ with the following properties: (i) The endomorphism algebra of $\mathbf{Q}$ in $K^b(\mathop{\rm proj}\nolimits B)$ is a factor algebra of $A$, and (ii) there are induced torsion pairs in $\mathop{\rm mod}\nolimits A$ and $\mathop{\rm mod}\nolimits B$, such that we obtain natural equivalences induced by $\mathop{\rm Hom}\nolimits$- and $\mathop{\rm Ext}\nolimits$-functors. Moreover, we show how the Auslander-Reiten theory of $\mathop{\rm mod}\nolimits B$ can be described in terms of the Auslander-Reiten theory of $\mathop{\rm mod}\nolimits A$.Open AccessA silting theorem
We give a generalization of the classical tilting theorem. We show that for a 2-term silting complex $\mathbf{P}$ in the bounded homotopy category $K^b(\mathop{\rm proj}\nolimits A)$ of finitely generated projective modules of a finite dimensional algebra $A$, the algebra $B = \mathop{\rm End}\nolimits_{K^b(\mathop{\rm proj}\nolimits A)}(\mathbf{P})$ admits a 2-term silting complex $\mathbf{Q}$ with the following properties: (i) The endomorphism algebra of $\mathbf{Q}$ in $K^b(\mathop{\rm proj}\nolimits B)$ is a factor algebra of $A$, and (ii) there are induced torsion pairs in $\mathop{\rm mod}\nolimits A$ and $\mathop{\rm mod}\nolimits B$, such that we obtain natural equivalences induced by $\mathop{\rm Hom}\nolimits$- and $\mathop{\rm Ext}\nolimits$-functors. Moreover, we show how the Auslander-Reiten theory of $\mathop{\rm mod}\nolimits B$ can be described in terms of the Auslander-Reiten theory of $\mathop{\rm mod}\nolimits A$. - Open Access
Algebras of finite representation type arising from maximal rigid objects
We give a complete classification of all algebras appearing as endomorphism algebras of maximal rigid objects in standard 2-Calabi-Yau categories of finite type. Such categories are equivalent to certain orbit categories of derived categories of Dynkin algebras. It turns out that with one exception, all the algebras that occur are $2$-Calabi-Yau-tilted, and therefore appear in an earlier classification by Bertani-{\O}kland and Oppermann. We explain this phenomenon by investigating the subcategories generated by rigid objects in standard 2-Calabi-Yau categories of finite type.
Open AccessAlgebras of finite representation type arising from maximal rigid objects
We give a complete classification of all algebras appearing as endomorphism algebras of maximal rigid objects in standard 2-Calabi-Yau categories of finite type. Such categories are equivalent to certain orbit categories of derived categories of Dynkin algebras. It turns out that with one exception, all the algebras that occur are $2$-Calabi-Yau-tilted, and therefore appear in an earlier classification by Bertani-{\O}kland and Oppermann. We explain this phenomenon by investigating the subcategories generated by rigid objects in standard 2-Calabi-Yau categories of finite type.
- Open Access
- 1 April 2013
- Published by Wiley in Journal of the London Mathematical Society
- Vol. 87 (2) , 643
From triangulated categories to module categories via localisation II: calculus of fractions
- 1 April 2013
- Published by Wiley in Journal of the London Mathematical Society
- Vol. 87 (2) , 643
Open AccessFrom triangulated categories to module categories via localisation II: calculus of fractions
- 20 March 2013
- Published by Springer Nature in Algebras and Representation Theory
- Vol. 17 (2) , 565-591
Torsion Pairs and Rigid Objects in Tubes
- 20 March 2013
- Published by Springer Nature in Algebras and Representation Theory
- Vol. 17 (2) , 565-591
Torsion Pairs and Rigid Objects in Tubes
- Open Access
Denominators in cluster algebras of affine type
The Fomin-Zelevinsky Laurent phenomenon states that every cluster variable in a cluster algebra can be expressed as a Laurent polynomial in the variables lying in an arbitrary initial cluster. We give representation-theoretic formulas for the denominators of cluster variables in cluster algebras of affine type. The formulas are in terms of the dimensions of spaces of homomorphisms in the corresponding cluster category, and hold for any choice of initial cluster.
Open AccessDenominators in cluster algebras of affine type
The Fomin-Zelevinsky Laurent phenomenon states that every cluster variable in a cluster algebra can be expressed as a Laurent polynomial in the variables lying in an arbitrary initial cluster. We give representation-theoretic formulas for the denominators of cluster variables in cluster algebras of affine type. The formulas are in terms of the dimensions of spaces of homomorphisms in the corresponding cluster category, and hold for any choice of initial cluster.
- Open Access
Torsion pairs and rigid objects in tubes
We classify the torsion pairs in a tube category and show that they are in bijection with maximal rigid objects in the extension of the tube category containing the Pruefer and adic modules. We show that the annulus geometric model for the tube category can be extended to the larger category and interpret torsion pairs, maximal rigid objects and the bijection between them geometrically. We also give a similar geometric description in the case of the linear orientation of a Dynkin quiver of type A.
Open AccessTorsion pairs and rigid objects in tubes
We classify the torsion pairs in a tube category and show that they are in bijection with maximal rigid objects in the extension of the tube category containing the Pruefer and adic modules. We show that the annulus geometric model for the tube category can be extended to the larger category and interpret torsion pairs, maximal rigid objects and the bijection between them geometrically. We also give a similar geometric description in the case of the linear orientation of a Dynkin quiver of type A.
- Open Access
- 7 November 2012
- Published by American Mathematical Society (AMS) in Transactions of the American Mathematical Society
- Vol. 365 (6) , 2845-2861
From triangulated categories to module categories via localisation
We show that the category of finite dimensional modules over the endomorphism algebra of a rigid object in a Hom-finite triangulated category is equivalent to the Gabriel-Zisman localisation of the category with respect to a certain class of maps. This generalises the 2 2 -Calabi-Yau tilting theorem of Keller-Reiten, in which the module category is obtained as a factor category, to the rigid case.
- 7 November 2012
- Published by American Mathematical Society (AMS) in Transactions of the American Mathematical Society
- Vol. 365 (6) , 2845-2861
Open AccessFrom triangulated categories to module categories via localisation
We show that the category of finite dimensional modules over the endomorphism algebra of a rigid object in a Hom-finite triangulated category is equivalent to the Gabriel-Zisman localisation of the category with respect to a certain class of maps. This generalises the 2 2 -Calabi-Yau tilting theorem of Keller-Reiten, in which the module category is obtained as a factor category, to the rigid case.