## Cluster algebras. I: Foundations.(English)Zbl 1021.16017

The authors introduce a new class of algebras called “cluster algebras”, and exhibit the cluster algebra structure for some well-known algebras such as $$\mathbb C[\text{SL}_3/N]$$ ($$N$$ being the subgroup of $$\text{SL}_3$$ consisting of all unipotent upper triangular matrices), $$\mathbb C[\text{Gr}_{2,n+3}]$$ ($$\text{Gr}_{2,n+3}$$ being the Grassmannian of 2-dimensional subspaces of $$\mathbb C^{n+3}$$). After introducing the cluster algebras, the authors first derive some structural properties of these algebras, then study the cluster algebras of rank 2. The authors also conjecture that for a complex semisimple, simply connected algebraic group $$G$$, $$\mathbb C[G]$$ and $$\mathbb C[G/N]$$ are cluster algebras.

### MSC:

 16S34 Group rings 13F60 Cluster algebras 16S50 Endomorphism rings; matrix rings 17B20 Simple, semisimple, reductive (super)algebras 20G05 Representation theory for linear algebraic groups 14M15 Grassmannians, Schubert varieties, flag manifolds
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### References:

 [1] Arkady Berenstein, Sergey Fomin, and Andrei Zelevinsky, Parametrizations of canonical bases and totally positive matrices, Adv. Math. 122 (1996), no. 1, 49 – 149. · Zbl 0966.17011 [2] Arkady Berenstein and Andrei Zelevinsky, String bases for quantum groups of type \?\?, I. M. Gel$$^{\prime}$$fand Seminar, Adv. Soviet Math., vol. 16, Amer. Math. Soc., Providence, RI, 1993, pp. 51 – 89. · Zbl 0794.17007 [3] Arkady Berenstein and Andrei Zelevinsky, Total positivity in Schubert varieties, Comment. Math. Helv. 72 (1997), no. 1, 128 – 166. · Zbl 0891.20030 [4] A. Berenstein and A. Zelevinsky, Tensor product multiplicities, canonical bases and totally positive varieties, Invent. Math. 143 (2001), 77-128. CMP 2001:06 · Zbl 1061.17006 [5] Sergey Fomin and Andrei Zelevinsky, Double Bruhat cells and total positivity, J. Amer. Math. Soc. 12 (1999), no. 2, 335 – 380. · Zbl 0913.22011 [6] Sergey Fomin and Andrei Zelevinsky, Total positivity: tests and parametrizations, Math. Intelligencer 22 (2000), no. 1, 23 – 33. · Zbl 1052.15500 [7] Sergey Fomin and Andrei Zelevinsky, Totally nonnegative and oscillatory elements in semisimple groups, Proc. Amer. Math. Soc. 128 (2000), no. 12, 3749 – 3759. · Zbl 0953.22016 [8] S. Fomin and A. Zelevinsky, The Laurent phenomenon, to appear in Adv. in Applied Math. · Zbl 1012.05012 [9] I. M. Gel$$^{\prime}$$fand and A. Zelevinsky, Canonical basis in irreducible representations of \?\?$$_{3}$$ and its applications, Group theoretical methods in physics, Vol. II (Yurmala, 1985) VNU Sci. Press, Utrecht, 1986, pp. 127 – 146. [10] Victor G. Kac, Infinite-dimensional Lie algebras, 3rd ed., Cambridge University Press, Cambridge, 1990. · Zbl 0716.17022 [11] 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. · Zbl 0577.15020 [12] 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. · Zbl 0894.14021 [13] G. Lusztig, Canonical bases arising from quantized enveloping algebras, J. Amer. Math. Soc. 3 (1990), no. 2, 447 – 498. · Zbl 0703.17008 [14] George Lusztig, Introduction to quantum groups, Progress in Mathematics, vol. 110, Birkhäuser Boston, Inc., Boston, MA, 1993. · Zbl 0788.17010 [15] G. Lusztig, Total positivity in reductive groups, Lie theory and geometry, Progr. Math., vol. 123, Birkhäuser Boston, Boston, MA, 1994, pp. 531 – 568. · Zbl 0845.20034 [16] A. V. Zelevinskiĭ and V. S. Retakh, The fundamental affine space and canonical basis in irreducible representations of the group \?\?$$_{4}$$, Dokl. Akad. Nauk SSSR 300 (1988), no. 1, 31 – 35 (Russian); English transl., Soviet Math. Dokl. 37 (1988), no. 3, 618 – 622. [17] 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. · Zbl 0998.20038 [18] Bernd Sturmfels, Algorithms in invariant theory, Texts and Monographs in Symbolic Computation, Springer-Verlag, Vienna, 1993. · Zbl 0802.13002 [19] Al. B. Zamolodchikov, On the thermodynamic Bethe ansatz equations for reflectionless \?\?\? scattering theories, Phys. Lett. B 253 (1991), no. 3-4, 391 – 394. [20] Andrei Zelevinsky, Connected components of real double Bruhat cells, Internat. Math. Res. Notices 21 (2000), 1131 – 1154. · Zbl 0978.20021
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