## Cluster algebras. III: Upper bounds and double Bruhat cells.(English)Zbl 1135.16013

Summary: We develop a new approach to cluster algebras, based on the notion of an upper cluster algebra defined as an intersection of Laurent polynomial rings. Strengthening the Laurent phenomenon established in part I [S. Fomin and A. Zelevinsky, J. Am. Math. Soc. 15, No. 2, 497-529 (2002; Zbl 1021.16017)], we show that under an assumption of “acyclicity”, a cluster algebra coincides with its upper counterpart and is finitely generated; in this case, we also describe its defining ideal and construct a standard monomial basis. We prove that the coordinate ring of any double Bruhat cell in a semisimple complex Lie group is naturally isomorphic to an upper cluster algebra explicitly defined in terms of relevant combinatorial data.

### MSC:

 16G20 Representations of quivers and partially ordered sets 16S15 Finite generation, finite presentability, normal forms (diamond lemma, term-rewriting) 17B20 Simple, semisimple, reductive (super)algebras 05E15 Combinatorial aspects of groups and algebras (MSC2010) 14M17 Homogeneous spaces and generalizations 22E46 Semisimple Lie groups and their representations 20G05 Representation theory for linear algebraic groups

### Citations:

Zbl 1021.16017; Zbl 1054.17024; Zbl 1127.16023
Full Text:

### References:

 [1] A. Berenstein and A. Zelevinsky, Tensor product multiplicities, canonical bases and totally positive varieties , Invent. Math. 143 (2001), 77–128. · Zbl 1061.17006 [2] N. Bourbaki, Éléments de mathématique, fasc. 34: Groupes et algèbres de Lie, chapitres 4–6 , Actualités Sci. Indust. 1337 , Hermann, Paris, 1968. [3] C. De Concini and C. Procesi, “Quantum Schubert cells and representations at roots of $$1$$” in Algebraic Groups and Lie Groups , Austral. Math. Soc. Lect. Ser. 9 , Cambridge Univ. Press, Cambridge, 1997, 127–160. · Zbl 0901.17005 [4] S. Fomin and A. Zelevinsky, Double Bruhat cells and total positivity , J. Amer. Math. Soc. 12 (1999), 335–380. JSTOR: · Zbl 0913.22011 [5] –. –. –. –., Recognizing Schubert cells , J. Algebraic Combin. 12 (2000), 37–57. · Zbl 1056.14504 [6] –. –. –. –., Total positivity: Tests and parametrizations , Math. Intelligencer 22 (2000), 23–33. · Zbl 1052.15500 [7] –. –. –. –., Cluster algebras I: Foundations , J. Amer. Math. Soc. 15 (2002), 497–529. JSTOR: · Zbl 1021.16017 [8] –. –. –. –., Cluster algebras II: Finite type classification , Invent. Math. 154 (2003), 63–121. · Zbl 1054.17024 [9] M. Gekhtman, M. Shapiro, and A. Vainshtein, Cluster algebras and Poisson geometry , Mosc. Math. J. 3 (2003), 899–934. · Zbl 1057.53064 [10] T. Hoffmann, J. Kellendonk, N. Kutz, and N. Reshetikhin, Factorization dynamics and Coxeter-Toda lattices , Comm. Math. Phys. 212 (2000), 297–321. · Zbl 0989.37074 [11] M. Kogan and A. Zelevinsky, On symplectic leaves and integrable systems in standard complex semisimple Poisson-Lie groups , Int. Math. Res. Not. 2002 , no. 32, 1685–1702. · Zbl 1006.22015 [12] V. Lakshmibai and C. S. Seshadri, Geometry of $$G/P$$, V , J. Algebra 100 (1986), 462–557. · Zbl 0618.14026 [13] B. Leclerc, Imaginary vectors in the dual canonical basis of $$U_q(\mathfrakn)$$ , Transform. Groups 8 (2003), 95–104. · Zbl 1044.17009 [14] G. Lusztig, “Total positivity in reductive groups” in Lie Theory and Geometry: In Honor of Bertram Kostant , Progr. Math. 123 , Birkhäuser, Boston, 1994, 531–568. · Zbl 0845.20034 [15] C. S. Seshadri, “Geometry of $$G/P$$, I: Theory of standard monomials for minuscule representations” in C. P. Ramanujam –.-A Tribute , Tata Inst. Fund. Res. Studies in Math. 8 , Springer, Berlin, 1978, 207–239. · Zbl 0447.14010 [16] B. Shapiro, M. Shapiro, A. Vainshtein, and A. Zelevinsky, Simply laced Coxeter groups and groups generated by symplectic transvections , Michigan Math. J. 48 (2000), 531–551. · Zbl 0998.20038 [17] A. Zelevinsky, Connected components of real double Bruhat cells , Int. Math. Res. Not. 2000 , no. 21, 1131–1153. · Zbl 0978.20021 [18] –. –. –. –., “From Littlewood-Richardson coefficients to cluster algebras in three lectures” in Symmetric Functions 2001: Surveys of Developments and Perspectives , NATO Sci. Ser. II Math. Phys. Chem. 74 , Kluwer, Dordrecht, Netherlands, 2002, 253–273. · Zbl 1155.17303
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.