Cluster algebras I: Foundations
Journal of the American Mathematical Society, Tome 15 (2002) no. 2, pp. 497-529

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

In an attempt to create an algebraic framework for dual canonical bases and total positivity in semisimple groups, we initiate the study of a new class of commutative algebras.
DOI : 10.1090/S0894-0347-01-00385-X

Fomin, Sergey 1 ; Zelevinsky, Andrei 2

1 Department of Mathematics, University of Michigan, Ann Arbor, Michigan 48109
2 Department of Mathematics, Northeastern University, Boston, Massachusetts 02115
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Fomin, Sergey; Zelevinsky, Andrei. Cluster algebras I: Foundations. Journal of the American Mathematical Society, Tome 15 (2002) no. 2, pp. 497-529. doi: 10.1090/S0894-0347-01-00385-X

[1] Berenstein, Arkady, Fomin, Sergey, Zelevinsky, Andrei Parametrizations of canonical bases and totally positive matrices Adv. Math. 1996 49 149

[2] Berenstein, Arkady, Zelevinsky, Andrei String bases for quantum groups of type 𝐴ᵣ 1993 51 89

[3] Berenstein, Arkady, Zelevinsky, Andrei Total positivity in Schubert varieties Comment. Math. Helv. 1997 128 166

[4] Fomin, Sergey, Zelevinsky, Andrei Double Bruhat cells and total positivity J. Amer. Math. Soc. 1999 335 380

[5] Fomin, Sergey, Zelevinsky, Andrei Total positivity: tests and parametrizations Math. Intelligencer 2000 23 33

[6] Fomin, Sergey, Zelevinsky, Andrei Totally nonnegative and oscillatory elements in semisimple groups Proc. Amer. Math. Soc. 2000 3749 3759

[7] Gel′Fand, I. M., Zelevinsky, A. Canonical basis in irreducible representations of 𝑔𝑙₃ and its applications 1986 127 146

[8] Kac, Victor G. Infinite-dimensional Lie algebras 1990

[9] Kung, Joseph P. S., Rota, Gian-Carlo The invariant theory of binary forms Bull. Amer. Math. Soc. (N.S.) 1984 27 85

[10] Leclerc, Bernard, Zelevinsky, Andrei Quasicommuting families of quantum Plücker coordinates 1998 85 108

[11] Lusztig, G. Canonical bases arising from quantized enveloping algebras J. Amer. Math. Soc. 1990 447 498

[12] Lusztig, George Introduction to quantum groups 1993

[13] Lusztig, G. Total positivity in reductive groups 1994 531 568

[14] Zelevinskiä­, A. V., Retakh, V. S. The fundamental affine space and canonical basis in irreducible representations of the group 𝑆𝑝₄ Dokl. Akad. Nauk SSSR 1988 31 35

[15] Shapiro, Boris, Shapiro, Michael, Vainshtein, Alek, Zelevinsky, Andrei Simply laced Coxeter groups and groups generated by symplectic transvections Michigan Math. J. 2000 531 551

[16] Sturmfels, Bernd Algorithms in invariant theory 1993

[17] Zamolodchikov, Al. B. On the thermodynamic Bethe ansatz equations for reflectionless 𝐴𝐷𝐸 scattering theories Phys. Lett. B 1991 391 394

[18] Zelevinsky, Andrei Connected components of real double Bruhat cells Internat. Math. Res. Notices 2000 1131 1154

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