Andrei Zelevinsky has main interests in representation theory, algebraic geometry, polyhedral and algebraic combinatorics and Lie theory (not in the sense as in $n$lab, but in the common sense of 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.
I. M. Gelfand, M. M. Kapranov, A. Zelevinsky, Discriminants, resultants, and multidimensional determinants, Birkhäuser 1994, 523 pp.
Sergey Fomin?, Andrei Zelevinsky, Cluster algebras. I. Foundations, J. Amer. Math. Soc. 15 (2002) no. 2, 497–529 math.RT/0104151; Cluster algebras. II. Finite type classifications. Invent. Math. 154 (2003) no. 1, 63–121 – classification of cluster algebras of finite type via root systems math.RA/0208229.
Arkady Berenstein, Andrei Zelevinsky, Quantum cluster algebras, math.QA/0404446
A. Zelevinsky’s homepage
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