Quantum Groups, Poisson Lie Groups, and Combinatorics

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This proposal addresses series of interrelated problems on quantum groups, Poisson Lie groups, Poisson homogeneous spaces, and their relations to combinatorics. These include: determining the spectra of quantized universal enveloping algebras of nilpotent algebras of De Concini-Kac-Procesi and Lusztig and the Berenstein-Zwicknagl (flat) braided symmetric algebras, carring out the orbit method for quantized algebras of functions, classifying automorphism groups of quantum nilpotent algebras, investigating quantized algebras of functions related to the Belavin-Drinfeld classification and the corresponding varieties of Lagrangian subalgebras, proving completeness of Kogan-Zelevinsky integrable systems, investigating cluster algebras on flag varieties and double flag varieties and their interaction with total positivity on these objects.The theory of Lie and algebraic groups is a major branch of mathematics, used to study in a unified way symmetries of physical theories, and geometric and algebraic objects. In recent years noncommutative versions of these techniques started playing an increasingly prominent role. Hopf algebras appeared in numerous studies in mathematical physics, algebra, combinatorics, noncommutative geometry, and topology.