QUANTISATION SPACES OF CLUSTER ALGEBRAS
Glasgow mathematical journal, Tome 60 (2018) no. 2, pp. 273-284

Voir la notice de l'article provenant de la source Cambridge University Press

The article concerns the existence and uniqueness of quantisations of cluster algebras. We prove that cluster algebras with an initial exchange matrix of full rank admit a quantisation in the sense of Berenstein-Zelevinsky and give an explicit generating set to construct all quantisations.
GELLERT, FLORIAN; LAMPE, PHILIPP. QUANTISATION SPACES OF CLUSTER ALGEBRAS. Glasgow mathematical journal, Tome 60 (2018) no. 2, pp. 273-284. doi: 10.1017/S0017089517000076
@article{10_1017_S0017089517000076,
     author = {GELLERT, FLORIAN and LAMPE, PHILIPP},
     title = {QUANTISATION {SPACES} {OF} {CLUSTER} {ALGEBRAS}},
     journal = {Glasgow mathematical journal},
     pages = {273--284},
     year = {2018},
     volume = {60},
     number = {2},
     doi = {10.1017/S0017089517000076},
     url = {http://geodesic.mathdoc.fr/articles/10.1017/S0017089517000076/}
}
TY  - JOUR
AU  - GELLERT, FLORIAN
AU  - LAMPE, PHILIPP
TI  - QUANTISATION SPACES OF CLUSTER ALGEBRAS
JO  - Glasgow mathematical journal
PY  - 2018
SP  - 273
EP  - 284
VL  - 60
IS  - 2
UR  - http://geodesic.mathdoc.fr/articles/10.1017/S0017089517000076/
DO  - 10.1017/S0017089517000076
ID  - 10_1017_S0017089517000076
ER  - 
%0 Journal Article
%A GELLERT, FLORIAN
%A LAMPE, PHILIPP
%T QUANTISATION SPACES OF CLUSTER ALGEBRAS
%J Glasgow mathematical journal
%D 2018
%P 273-284
%V 60
%N 2
%U http://geodesic.mathdoc.fr/articles/10.1017/S0017089517000076/
%R 10.1017/S0017089517000076
%F 10_1017_S0017089517000076

[1] 1. Berenstein, A., Fomin, S. and Zelevinsky, A., Cluster algebras. III. Upper bounds and double Bruhat cells, Duke Math. J. 126 (1) (2005), 1–52. Google Scholar

[2] 2. Berenstein, A. and Zelevinsky, A., Quantum cluster algebras, Adv. Math. 195 (2) (2005), 405–455. Google Scholar | DOI

[3] 3. Burban, I., Iyama, O., Keller, B. and Reiten, I., Cluster tilting for one-dimensional hypersurface singularities. Adv. Math. 217 (6) (2008), 2443–2484. Google Scholar | DOI

[4] 4. Caldero, P. and Chapoton, F., Cluster algebras as Hall algebras of quiver representations, Comment. Math. Helv. 81 (3) (2006), 595–616. Google Scholar

[5] 5. Cayley, A., Sur les déterminants gauches, J. Reine Angew. Math. 38 (1849), 93–96. Google Scholar

[6] 6. Fock, V. V. and Goncharov, A. B., Cluster ensembles, quantization and the dilogarithm, Ann. Sci. École Norm. Sup., série 4, 42 (6) (2009), 865–930. Google Scholar | DOI

[7] 7. Fomin, S., Shapiro, M. and Thurston, D., Cluster algebras and triangulated surfaces, I. Cluster complexes, Acta Math. 201 (1) (2008), 83–146. Google Scholar | DOI

[8] 8. Fomin, S. and Thurston, D., Cluster algebras and triangulated surfaces. Part II: Lambda lengths. Preprint: arXiv 1210.5569 (2012). To appear in Memoirs of Amer. Math. Soc. Google Scholar

[9] 9. Fomin, S. and Zelevinsky, A., Cluster algebras. I. Foundations. J. Amer. Math. Soc. 15 (2) (2002), 497–529 (electronic). Google Scholar

[10] 10. Fomin, S. and Zelevinsky, A., Cluster algebras. II. Finite type classification, Invent. Math. 154 (1) (2003), 63–121. Google Scholar

[11] 11. Fomin, S. and Zelevinsky, A., Cluster algebras. IV. Coefficients, Compos. Math. 143 (1) (2007), 112–164. Google Scholar

[12] 12. Geiß, C., Leclerc, B. and Schröer, J., Cluster structures on quantum coordinate rings, Selecta Math. (N.S.) 19 (2) (2013), 337–397. Google Scholar

[13] 13. Gekhtman, M., Shapiro, M. and Vainshtein, A., Cluster algebras and Poisson geometry, Mosc. Math. J. 3 (3) (2003), 899–934. Google Scholar | DOI

[14] 14. Gellert, F., Sage functions for the quantisation of cluster algebras. http://math.uni-bielefeld.de/~fgellert/quantisation.php Google Scholar

[15] 15. Goodearl, K. R. and Yakimov, M. T., Quantum cluster algebra structures on quantum nilpotent algebras. Mem. Amer. Math. Soc. 247 (1169) (2017), vii+119 pp. Google Scholar

[16] 16. Grabowski, J. E., Graded cluster algebras, J. Algebr. Combin. 42 (4) (2015), 1111–1134. Google Scholar | DOI

[17] 17. Grabowski, J. E. and Launois, S., Graded quantum cluster algebras and an application to quantum grassmannians, Proc. London Math. Soc. 109 (3) (2014), 697–732. Google Scholar | DOI

[18] 18. Hernandez, D. and Leclerc, B., Cluster algebras and quantum affine algebras, Duke Math. J. 154 (2) (2010), 265–341. Google Scholar

[19] 19. Knuth, D. E., Overlapping Pfaffians, Electron. J. Combin. 3 (2), Research Paper 5 (1996). Google Scholar

[20] 20. Kimura, Y. and Qin, F., Graded quiver varieties, quantum cluster algebras and dual canonical basis, Adv. Math. 262 (2014), 261–312. Google Scholar | DOI

[21] 21. Lampe, P., A quantum cluster algebra of Kronecker type and the dual canonical basis, Int. Math. Res. Not. IMRN 2011 (13) (2011), 2970–3005. Google Scholar

[22] 22. Lampe, P., Quantum cluster algebras of type A and the dual canonical basis, Proc. London Math. Soc. 108 (2014), 1–43. Google Scholar

[23] 23. Leclerc, B., Dual canonical bases, quantum shuffles and q-characters, Math. Z. 246 (4) (2004), 691–732. Google Scholar

[24] 24. Lusztig, G., Introduction to quantum groups, Progress in mathematics, vol. 110 (Birkhäuser Boston Inc., Boston, MA, 1993). Google Scholar

[25] 25. Rupel, D., The Feigin tetrahedron, SIGMA 11 (024) (2015), 30 pages. Google Scholar

[26] 26. Zelevinsky, A., Quantum cluster algebras, Lecture, Infinite Analysis, vol. 11 (Winter School, Osaka University, Japan, 2011). Google Scholar

Cité par Sources :