Cluster algebras I: Foundations

Fomin, Sergey; Zelevinsky, Andrei

doi:10.1090/S0894-0347-01-00385-X

Cluster algebras I: Foundations
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by Sergey Fomin and Andrei Zelevinsky;

J. Amer. Math. Soc. 15 (2002), 497-529

DOI: https://doi.org/10.1090/S0894-0347-01-00385-X

Published electronically: December 28, 2001

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Abstract:

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.

References

Arkady Berenstein, Sergey Fomin, and Andrei Zelevinsky, Parametrizations of canonical bases and totally positive matrices, Adv. Math. 122 (1996), no. 1, 49–149. MR 1405449, DOI 10.1006/aima.1996.0057
Arkady Berenstein and Andrei Zelevinsky, String bases for quantum groups of type $A_r$, I. M. Gel′fand Seminar, Adv. Soviet Math., vol. 16, Amer. Math. Soc., Providence, RI, 1993, pp. 51–89. MR 1237826
Arkady Berenstein and Andrei Zelevinsky, Total positivity in Schubert varieties, Comment. Math. Helv. 72 (1997), no. 1, 128–166. MR 1456321, DOI 10.1007/PL00000363

Invent. Math.

143

Sergey Fomin and Andrei Zelevinsky, Double Bruhat cells and total positivity, J. Amer. Math. Soc. 12 (1999), no. 2, 335–380. MR 1652878, DOI 10.1090/S0894-0347-99-00295-7
Sergey Fomin and Andrei Zelevinsky, Total positivity: tests and parametrizations, Math. Intelligencer 22 (2000), no. 1, 23–33. MR 1745560, DOI 10.1007/BF03024444
Sergey Fomin and Andrei Zelevinsky, Totally nonnegative and oscillatory elements in semisimple groups, Proc. Amer. Math. Soc. 128 (2000), no. 12, 3749–3759. MR 1694341, DOI 10.1090/S0002-9939-00-05487-3

Adv. in Applied Math.

I. M. Gel′fand and A. Zelevinsky, Canonical basis in irreducible representations of $\textrm {gl}_3$ and its applications, Group theoretical methods in physics, Vol. II (Yurmala, 1985) VNU Sci. Press, Utrecht, 1986, pp. 127–146. MR 919787
Victor G. Kac, Infinite-dimensional Lie algebras, 3rd ed., Cambridge University Press, Cambridge, 1990. MR 1104219, DOI 10.1017/CBO9780511626234
Joseph P. S. Kung and Gian-Carlo Rota, The invariant theory of binary forms, Bull. Amer. Math. Soc. (N.S.) 10 (1984), no. 1, 27–85. MR 722856, DOI 10.1090/S0273-0979-1984-15188-7
Bernard Leclerc and Andrei Zelevinsky, Quasicommuting families of quantum Plücker coordinates, Kirillov’s seminar on representation theory, Amer. Math. Soc. Transl. Ser. 2, vol. 181, Amer. Math. Soc., Providence, RI, 1998, pp. 85–108. MR 1618743, DOI 10.1090/trans2/181/03
G. Lusztig, Canonical bases arising from quantized enveloping algebras, J. Amer. Math. Soc. 3 (1990), no. 2, 447–498. MR 1035415, DOI 10.1090/S0894-0347-1990-1035415-6
George Lusztig, Introduction to quantum groups, Progress in Mathematics, vol. 110, Birkhäuser Boston, Inc., Boston, MA, 1993. MR 1227098
G. Lusztig, Total positivity in reductive groups, Lie theory and geometry, Progr. Math., vol. 123, Birkhäuser Boston, Boston, MA, 1994, pp. 531–568. MR 1327548, DOI 10.1007/978-1-4612-0261-5_{2}0
A. V. Zelevinskiĭ and V. S. Retakh, The fundamental affine space and canonical basis in irreducible representations of the group $\textrm {Sp}_4$, Dokl. Akad. Nauk SSSR 300 (1988), no. 1, 31–35 (Russian); English transl., Soviet Math. Dokl. 37 (1988), no. 3, 618–622. MR 948800
Boris Shapiro, Michael Shapiro, Alek Vainshtein, and Andrei Zelevinsky, Simply laced Coxeter groups and groups generated by symplectic transvections, Michigan Math. J. 48 (2000), 531–551. Dedicated to William Fulton on the occasion of his 60th birthday. MR 1786504, DOI 10.1307/mmj/1030132732
Bernd Sturmfels, Algorithms in invariant theory, Texts and Monographs in Symbolic Computation, Springer-Verlag, Vienna, 1993. MR 1255980, DOI 10.1007/978-3-7091-4368-1
Al. B. Zamolodchikov, On the thermodynamic Bethe ansatz equations for reflectionless $ADE$ scattering theories, Phys. Lett. B 253 (1991), no. 3-4, 391–394. MR 1092210, DOI 10.1016/0370-2693(91)91737-G
Andrei Zelevinsky, Connected components of real double Bruhat cells, Internat. Math. Res. Notices 21 (2000), 1131–1154. MR 1800992, DOI 10.1155/S1073792800000568