Cluster $\mathcal X$-varieties for dual Poisson–Lie groups. I
HTML articles powered by AMS MathViewer
- by R. Brahami
- St. Petersburg Math. J. 22 (2011), 183-250
- DOI: https://doi.org/10.1090/S1061-0022-2011-01138-0
- Published electronically: February 8, 2011
- PDF | Request permission
Abstract:
We associate a family of cluster $\mathcal X$-varieties with the dual Poisson–Lie group $G^*$ of a complex semisimple Lie group $G$ of adjoint type given with the standard Poisson structure. This family is described by the $W$-permutohedron associated with the Lie algebra $\mathfrak g$ of $G$, vertices being labeled by cluster $\mathcal X$-varieties and edges by new Poisson birational isomorphisms on appropriate seed $\mathcal X$-tori, called saltation. The underlying combinatorics is based on a factorization of the Fomin–Zelevinsky twist maps into mutations and other new Poisson birational isomorphisms on seed $\mathcal X$-tori, called tropical mutations (because they are obtained by a tropicalization of the mutation formula), associated with an enrichment of the combinatorics on double words of the Weyl group $W$ of $G$.References
- A. Yu. Alekseev and A. Z. Malkin, Symplectic structures associated to Lie-Poisson groups, Comm. Math. Phys. 162 (1994), no. 1, 147–173. MR 1272770
- Arkady Berenstein, Sergey Fomin, and Andrei Zelevinsky, Cluster algebras. III. Upper bounds and double Bruhat cells, Duke Math. J. 126 (2005), no. 1, 1–52. MR 2110627, DOI 10.1215/S0012-7094-04-12611-9
- Arkady Berenstein and Andrei Zelevinsky, Tensor product multiplicities, canonical bases and totally positive varieties, Invent. Math. 143 (2001), no. 1, 77–128. MR 1802793, DOI 10.1007/s002220000102
- Anders Björner and Francesco Brenti, Combinatorics of Coxeter groups, Graduate Texts in Mathematics, vol. 231, Springer, New York, 2005. MR 2133266
- Renaud Brahami, Cluster $\mathcal X$-varieties and dual Poisson–Lie groups. II (in preparation).
- Sam Evens and Jiang-Hua Lu, Poisson geometry of the Grothendieck resolution of a complex semisimple group, Mosc. Math. J. 7 (2007), no. 4, 613–642, 766 (English, with English and Russian summaries). MR 2372206, DOI 10.17323/1609-4514-2007-7-4-613-642
- Vladimir Fock and Alexander Goncharov, Moduli spaces of local systems and higher Teichmüller theory, Publ. Math. Inst. Hautes Études Sci. 103 (2006), 1–211. MR 2233852, DOI 10.1007/s10240-006-0039-4
- V. V. Fock and A. B. Goncharov, Cluster $\scr X$-varieties, amalgamation, and Poisson-Lie groups, Algebraic geometry and number theory, Progr. Math., vol. 253, Birkhäuser Boston, Boston, MA, 2006, pp. 27–68. MR 2263192, DOI 10.1007/978-0-8176-4532-8_{2}
- —, Cluster ensembles, quantization and the dilogarithm, arXiv:math/0311245.
- —, The quantum dilogarithm and unitary representations quantized cluster varieties, arXiv:math/0702397.
- Sergey Fomin and Nathan Reading, Root systems and generalized associahedra, Geometric combinatorics, IAS/Park City Math. Ser., vol. 13, Amer. Math. Soc., Providence, RI, 2007, pp. 63–131. MR 2383126, DOI 10.1090/pcms/013/03
- Sergey Fomin, Michael Shapiro, and Dylan Thurston, Cluster algebras and triangulated surfaces. Pt. I: Cluster complexes, arXiv:math/0608367.
- Sergey Fomin and Andrei Zelevinsky, Cluster algebras. I. Foundations, J. Amer. Math. Soc. 15 (2002), no. 2, 497–529. MR 1887642, DOI 10.1090/S0894-0347-01-00385-X
- Sergey Fomin and Andrei Zelevinsky, Cluster algebras. II. Finite type classification, Invent. Math. 154 (2003), no. 1, 63–121. MR 2004457, DOI 10.1007/s00222-003-0302-y
- Sergey Fomin and Andrei Zelevinsky, Cluster algebras. IV. Coefficients, Compos. Math. 143 (2007), no. 1, 112–164. MR 2295199, DOI 10.1112/S0010437X06002521
- 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
- William Fulton and Joe Harris, Representation theory, Graduate Texts in Mathematics, vol. 129, Springer-Verlag, New York, 1991. A first course; Readings in Mathematics. MR 1153249, DOI 10.1007/978-1-4612-0979-9
- Michael Gekhtman, Michael Shapiro, and Alek Vainshtein, Cluster algebras and Poisson geometry, Mosc. Math. J. 3 (2003), no. 3, 899–934, 1199 (English, with English and Russian summaries). {Dedicated to Vladimir Igorevich Arnold on the occasion of his 65th birthday}. MR 2078567, DOI 10.17323/1609-4514-2003-3-3-899-934
- Tim Hoffmann, Johannes Kellendonk, Nadja Kutz, and Nicolai Reshetikhin, Factorization dynamics and Coxeter-Toda lattices, Comm. Math. Phys. 212 (2000), no. 2, 297–321. MR 1772248, DOI 10.1007/s002200000212
- Mikhail Kogan and Andrei Zelevinsky, On symplectic leaves and integrable systems in standard complex semisimple Poisson-Lie groups, Int. Math. Res. Not. 32 (2002), 1685–1702. MR 1916837, DOI 10.1155/S1073792802203050
- Jiang-Hua Lu and Alan Weinstein, Poisson Lie groups, dressing transformations, and Bruhat decompositions, J. Differential Geom. 31 (1990), no. 2, 501–526. MR 1037412
- Alexander Postnikov, Permutohedra, associahedra, and beyond, Int. Math. Res. Not. IMRN 6 (2009), 1026–1106. MR 2487491, DOI 10.1093/imrn/rnn153
- N. Reshetikhin, Integrability of characteristic Hamiltonian systems on simple Lie groups with standard Poisson Lie structure, Comm. Math. Phys. 242 (2003), no. 1-2, 1–29. MR 2018267, DOI 10.1007/s00220-003-0916-3
- Michael A. Semenov-Tian-Shansky, Dressing transformations and Poisson group actions, Publ. Res. Inst. Math. Sci. 21 (1985), no. 6, 1237–1260. MR 842417, DOI 10.2977/prims/1195178514
- M. A. Semenov-Tyan-ShanskiÄ, Poisson-Lie groups. The quantum duality principle and the twisted quantum double, Teoret. Mat. Fiz. 93 (1992), no. 2, 302–329 (Russian, with English and Russian summaries); English transl., Theoret. and Math. Phys. 93 (1992), no. 2, 1292–1307 (1993). MR 1233548, DOI 10.1007/BF01083527
Bibliographic Information
- R. Brahami
- Affiliation: Institut Mathématique de Bourgogne, Dijon, France
- Email: Renaud.Brahami@u-bourgogne.fr
- Received by editor(s): September 22, 2009
- Published electronically: February 8, 2011
- © Copyright 2011 American Mathematical Society
- Journal: St. Petersburg Math. J. 22 (2011), 183-250
- MSC (2010): Primary 22E70, 81R10
- DOI: https://doi.org/10.1090/S1061-0022-2011-01138-0
- MathSciNet review: 2668124