×

zbMATH — the first resource for mathematics

Generalized cluster structure on the Drinfeld double of \(\mathrm{GL}_n\). (Structures d’algébres amassées généralisées sur le double de Drinfeld du group \(\mathrm{GL}_n\).) (English. French summary) Zbl 1387.13047
Summary: We construct a generalized cluster structure compatible with the Poisson bracket on the Drinfeld double of the standard Poisson-Lie group \(\mathrm{GL}_n\) and derive from it a generalized cluster structure in \(\mathrm{GL}_n\) compatible with the push-forward of the dual Poisson-Lie bracket.

MSC:
13F60 Cluster algebras
17B62 Lie bialgebras; Lie coalgebras
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Belavin, A.; Drinfeld, V., Solutions of the classical Yang-Baxter equation for simple Lie algebras, Funkc. Anal. Prilozh., 16, 1-29, (1982)
[2] Berenstein, A.; Fomin, S.; Zelevinsky, A., Cluster algebras. III. upper bounds and double Bruhat cells, Duke Math. J., 126, 1-52, (2005) · Zbl 1135.16013
[3] Brahami, R., Cluster χ-varieties for dual Poisson-Lie groups. I, Algebra Anal., 22, 14-104, (2010)
[4] Chekhov, L.; Shapiro, M., Teichmüller spaces of Riemann surfaces with orbifold points of arbitrary order and cluster variables, Int. Math. Res. Not., 2014, 10, 2746-2772, (2014) · Zbl 1301.30042
[5] Gekhtman, M.; Shapiro, M.; Vainshtein, A., Cluster algebras and Poisson geometry, Mosc. Math. J., 3, 899-934, (2003) · Zbl 1057.53064
[6] Gekhtman, M.; Shapiro, M.; Vainshtein, A., Cluster algebras and Poisson geometry, Mathematical Surveys and Monographs, vol. 167, (2010), The American Mathematical Society Providence, RI, USA · Zbl 1217.13001
[7] Gekhtman, M.; Shapiro, M.; Vainshtein, A., Cluster structures on simple complex Lie groups and Belavin-Drinfeld classification, Mosc. Math. J., 12, 293-312, (2012) · Zbl 1259.53075
[8] Gekhtman, M.; Shapiro, M.; Vainshtein, A., Cremmer-gervais cluster structure on \(\mathit{SL}_n\), Proc. Natl. Acad. Sci. USA, 111, 27, 9688-9695, (2014) · Zbl 1355.17023
[9] Gekhtman, M.; Shapiro, M.; Vainshtein, A., Exotic cluster structures on \(\mathit{SL}_n\): the Cremmer-gervais case, Mem. Amer. Math. Soc., (2016), in press · Zbl 1373.53112
[10] Reyman, A.; Semenov-Tian-Shansky, M., Group-theoretical methods in the theory of finite-dimensional integrable systems, (Encyclopaedia of Mathematical Sciences, vol. 16, (1994), Springer-Verlag Berlin), 116-225
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.