Andrei Zelevinsky’s main interests were in representation theory, algebraic geometry, polyhedral and algebraic combinatorics and Lie theory (the circle of questions relating Lie algebras, Lie groups, Kac-Moody groups, quantum groups etc.).
With Sergey Fomin, he created a theory of cluster algebras. He studied multidimensional generalizations of hypergeometric functions, arrangements of hyperplanes, bases for quantum groups, quantum determinants and minors, Grassmanians, flag and Schubert varieties etc.
Introducing cluster algebras:
Sergey Fomin, Andrei Zelevinsky: Cluster algebras I: Foundations, J. Amer. Math. Soc. 15 2 (2002) 497-529 [math.RT/0104151, doi:10.1090/S0894-0347-01-00385-X]
Sergey Fomin, Andrei Zelevinsky: Cluster algebras II: Finite type classifications, Invent. Math. 154 1 (2003) 63-121 [doi:10.1007/s00222-003-0302-y, arXiv:math/0208229]
Arkady Berenstein, Sergey Fomin, Andrei Zelevinsky: Cluster algebras III: Upper bounds and double Bruhat cells, Duke Math. J. 126 1 (2005) 1-52 [doi:10.1215/S0012-7094-04-12611-9, arXiv:math/0305434]
Sergey Fomin, Andrei Zelevinsky: Cluster algebras IV: Coefficients, Compos. Math. 143 (2007) 112-164 [doi:10.1112/S0010437X06002521, arXiv:math/0602259, MR2295199]
See also:
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