Quantum cluster algebras. (English) Zbl 1124.20028

The authors introduce and study quantum deformations of cluster algebras. The latter form a family of commutative rings which can be used in the theory of semisimple groups. To deform such algebras can be useful in Teichmüller theory. After recalling the relevant facts about cluster algebras, the authors construct and study the quantum cluster algebras. Examples in relation with Bruhat cells are also given.


20G05 Representation theory for linear algebraic groups
17B37 Quantum groups (quantized enveloping algebras) and related deformations
16G20 Representations of quivers and partially ordered sets
16S80 Deformations of associative rings
20G42 Quantum groups (quantized function algebras) and their representations
14M17 Homogeneous spaces and generalizations
22E46 Semisimple Lie groups and their representations
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[1] A. Berenstein, Group-like elements in quantum groups and Feigin’s conjecture, J. Algebra, to appear.
[2] A. Berenstein, S. Fomin, A. Zelevinsky, Cluster algebras III: upper bounds and double Bruhat cells, Duke Math. J., to appear. · Zbl 1135.16013
[3] Brown, K.; Goodearl, K., Lectures on algebraic quantum groups, (2002), Birkhäuser Basel · Zbl 1027.17010
[4] C. De Concini, C. Procesi Quantum Schubert cells and representations at roots of 1, in: Algebraic Groups and Lie Groups, Australian Mathematical Society Lecture Series, vol. 9, Cambridge University Press, Cambridge, 1997, pp. 127-160. · Zbl 0901.17005
[5] V.V. Fock, A.B. Goncharov, Moduli spaces of local systems and higher Teichmuller theory, . · Zbl 1099.14025
[6] V.V. Fock, A.B. Goncharov, Cluster ensembles, quantization and the dilogarithm, . · Zbl 1225.53070
[7] Fomin, S.; Zelevinsky, A., Double Bruhat cells and total positivity, J. amer. math. soc., 12, 335-380, (1999) · Zbl 0913.22011
[8] Fomin, S.; Zelevinsky, A., Cluster algebras ifoundations, J. amer. math. soc., 15, 497-529, (2002) · Zbl 1021.16017
[9] Fomin, S.; Zelevinsky, A., The Laurent phenomenon, Adv. appl. math., 28, 119-144, (2002) · Zbl 1012.05012
[10] Fomin, S.; Zelevinsky, A., Cluster algebras iifinite type classification, Invent. math., 154, 63-121, (2003) · Zbl 1054.17024
[11] Gekhtman, M.; Shapiro, M.; Vainshtein, A., Cluster algebras and Poisson geometry, Moscow math. J., 3, 3, (2003) · Zbl 1057.53064
[12] M. Gekhtman, M. Shapiro, A. Vainshtein, Cluster algebras and Weil-Petersson forms, . · Zbl 1079.53124
[13] Iohara, K.; Malikov, F., Rings of skew polynomials and Gel’fand-Kirillov conjecture for quantum groups, Comm. math. phys., 164, 217-238, (1994) · Zbl 0826.17011
[14] A. Joseph, Quantum Groups and their Primitive Ideals, Ergebnisse der Math. (3), vol. 29, Springer, Berlin, 1995. · Zbl 0808.17004
[15] Kac, V., Infinite dimensional Lie algebras, (1990), Cambridge University Press Cambridge · Zbl 0716.17022
[16] Kogan, M.; Zelevinsky, A., On symplectic leaves and integrable systems in standard complex semisimple Poisson-Lie groups, Internat. math. res. notices, 32, 1685-1702, (2002) · Zbl 1006.22015
[17] L. Korogodski, Y. Soibelman, Algebras of Functions on Quantum Groups, Part I. Mathematical Surveys and Monographs, vol. 56, American Mathematical Society, Providence, RI, 1998. · Zbl 0923.17017
[18] G. Lusztig, Introduction to Quantum Groups, Progress in Mathematics, vol. 110, Birkhäuser Boston, 1993. · Zbl 0788.17010
[19] G. Lusztig, Problems on canonical bases, in: W.J. Haboush, B.J. Parshall (Eds.), Algebraic groups and their generalizations: quantum and infinite-dimensional methods (University Park, PA, 1991), Proceedings of the Symposium on Pure Math., vol. 56, Part 2, American Mathematical Society, Providence, RI, 1994, pp. 169-176.
[20] P. Sherman, A. Zelevinsky, Positivity and canonical bases in rank 2 cluster algebras of finite and affine types, Moscow Math. J., to appear. · Zbl 1103.16018
[21] Zelevinsky, A., Connected components of real double Bruhat cells, Internat. math. res. notices, 21, 1131-1153, (2000) · Zbl 0978.20021
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