Geodesic
Quivers with potentials and their representations II: Applications to cluster algebras
Derksen, Harm1 ; Weyman, Jerzy2 ; Zelevinsky, Andrei2
1 Department of Mathematics, University of Michigan, Ann Arbor, Michigan 48109
2 Department of Mathematics, Northeastern University, Boston, Massachusetts 02115
Journal of the American Mathematical Society, Tome 23 (2010) no. 3, pp. 749-790

Voir la notice de l'article provenant de la source American Mathematical Society

We continue the study of quivers with potentials and their representations initiated in the first paper of the series. Here we develop some applications of this theory to cluster algebras. As shown in the “Cluster algebras IV” paper, the cluster algebra structure is to a large extent controlled by a family of integer vectors called $\mathbf {g}$-vectors, and a family of integer polynomials called $F$-polynomials. In the case of skew-symmetric exchange matrices we find an interpretation of these $\mathbf {g}$-vectors and $F$-polynomials in terms of (decorated) representations of quivers with potentials. Using this interpretation, we prove most of the conjectures about $\mathbf {g}$-vectors and $F$-polynomials made in loc. cit.
DOI : 10.1090/S0894-0347-10-00662-4

Derksen, Harm 1 ; Weyman, Jerzy 2 ; Zelevinsky, Andrei 2

1 Department of Mathematics, University of Michigan, Ann Arbor, Michigan 48109
2 Department of Mathematics, Northeastern University, Boston, Massachusetts 02115 
@article{10_1090_S0894_0347_10_00662_4,
     author = {Derksen, Harm and Weyman, Jerzy and Zelevinsky, Andrei},
     title = {Quivers with potentials and their representations {II:} {Applications} to cluster algebras},
     journal = {Journal of the American Mathematical Society},
     pages = {749--790},
     publisher = {mathdoc},
     volume = {23},
     number = {3},
     year = {2010},
     doi = {10.1090/S0894-0347-10-00662-4},
     url = {http://geodesic.mathdoc.fr/articles/10.1090/S0894-0347-10-00662-4/}
}
TY  - JOUR
AU  - Derksen, Harm
AU  - Weyman, Jerzy
AU  - Zelevinsky, Andrei
TI  - Quivers with potentials and their representations II: Applications to cluster algebras
JO  - Journal of the American Mathematical Society
PY  - 2010
SP  - 749
EP  - 790
VL  - 23
IS  - 3
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/articles/10.1090/S0894-0347-10-00662-4/
DO  - 10.1090/S0894-0347-10-00662-4
ID  - 10_1090_S0894_0347_10_00662_4
ER  - 
%0 Journal Article
%A Derksen, Harm
%A Weyman, Jerzy
%A Zelevinsky, Andrei
%T Quivers with potentials and their representations II: Applications to cluster algebras
%J Journal of the American Mathematical Society
%D 2010
%P 749-790
%V 23
%N 3
%I mathdoc
%U http://geodesic.mathdoc.fr/articles/10.1090/S0894-0347-10-00662-4/
%R 10.1090/S0894-0347-10-00662-4
%F 10_1090_S0894_0347_10_00662_4
Derksen, Harm; Weyman, Jerzy; Zelevinsky, Andrei. Quivers with potentials and their representations II: Applications to cluster algebras. Journal of the American Mathematical Society, Tome 23 (2010) no. 3, pp. 749-790. doi: 10.1090/S0894-0347-10-00662-4

[1] Assem, Ibrahim, Simson, Daniel, Skowroå„Ski, Andrzej Elements of the representation theory of associative algebras. Vol. 1 2006

[2] Biaå‚Ynicki-Birula, A. On fixed point schemes of actions of multiplicative and additive groups Topology 1973 99 103

[3] Buan, Aslak Bakke, Marsh, Bethany R., Reiten, Idun Denominators of cluster variables J. Lond. Math. Soc. (2) 2009 589 611

[4] Butler, M. C. R., King, A. D. Minimal resolutions of algebras J. Algebra 1999 323 362

[5] Caldero, Philippe, Chapoton, Frã©Dã©Ric Cluster algebras as Hall algebras of quiver representations Comment. Math. Helv. 2006 595 616

[6] Caldero, Philippe, Keller, Bernhard From triangulated categories to cluster algebras. II Ann. Sci. École Norm. Sup. (4) 2006 983 1009

[7] Caldero, Philippe, Reineke, Markus On the quiver Grassmannian in the acyclic case J. Pure Appl. Algebra 2008 2369 2380

[8] Derksen, Harm, Weyman, Jerzy, Zelevinsky, Andrei Quivers with potentials and their representations. I. Mutations Selecta Math. (N.S.) 2008 59 119

[9] Fomin, Sergey, Zelevinsky, Andrei Cluster algebras. I. Foundations J. Amer. Math. Soc. 2002 497 529

[10] Fomin, Sergey, Zelevinsky, Andrei Cluster algebras. IV. Coefficients Compos. Math. 2007 112 164

[11] Fu, Changjian, Keller, Bernhard On cluster algebras with coefficients and 2-Calabi-Yau categories Trans. Amer. Math. Soc. 2010 859 895

[12] Fulton, William Introduction to toric varieties 1993

[13] Marsh, Robert, Reineke, Markus, Zelevinsky, Andrei Generalized associahedra via quiver representations Trans. Amer. Math. Soc. 2003 4171 4186

[14] Schofield, Aidan General representations of quivers Proc. London Math. Soc. (3) 1992 46 64

[15] Schubert, Horst Categories 1972

[16] Shafarevich, Igor R. Basic algebraic geometry. 1 1994

Cité par Sources :