Mathematics – Rings and Algebras
Scientific paper
2009-11-15
Mathematics
Rings and Algebras
S\'eminaire Bourbaki, 62e ann\'ee, 2009-2010, expos\'e 1014, novembre 2009. 26 pages, in French
Scientific paper
Sergey Fomin and Andrei Zelevinsky have invented cluster algebras at the beginning of this decade with the aim of creating an algebraic framework for the study of canonical bases in quantum groups and total positivity in algebraic groups. It soon turned out that the combinatorics of cluster algebras also appear in many other subjects and notably in the representation theory of quivers and finite-dimensional algebras. In this talk, we give a concise introduction to cluster algebras and sketch two significant applications concerning the study of certain discrete dynamical systems and the construction of dual semicanonical bases. ----- Sergey Fomin et Andrei Zelevinsky ont invent\'e les alg\`ebres amass\'ees (cluster algebras) au d\'ebut des ann\'ees 2000 dans le but de fournir un cadre alg\'ebrique \`a l'\'etude des bases canoniques dans les groupes quantiques et de la positivit\'e totale dans les groupes alg\'ebriques. Il s'est av\'er\'e rapidement que la combinatoire des alg\`ebres amass\'ees intervenait \'egalement dans de nombreux autres sujets et notamment en th\'eorie des repr\'esentations des carquois et des alg\`ebres de dimension finie. Dans cet expos\'e, nous donnons une introduction concise aux alg\`ebres amass\'ees et esquissons deux applications significatives portant sur l'\'etude de certains syst\`emes dynamiques discrets et la construction de bases duales semi-canoniques.
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