Representation Theory, Cluster algebras, and Canonical Bases

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The PI plans to conduct research in an area of mathematics at the interface of the representation theory of Lie algebras and quantum groups, cluster algebras, and noncommutative algebraic geometry. The representation theory of Lie algebras and quantum groups is a dynamically developing field of modern mathematics. It has had a large impact in other areas of mathematics, as well as in physics. A central theme of this proposal is to understand and investigate the properties of canonical and crystal bases, which endow certain fundamental algebras (such as quantized enveloping algebras) with an additional structure that has proven to be fruitful for understanding these algebras and their realizations as matrices. Understanding the relationship between these bases and closely related structures, such as totally positive varieties, geometric crystals, and cluster algebras, and their quantum and totally noncommutative analogues, is a unifying theme of this proposal.In one project the PI proposes a new approach to the study of Hecke algebras based on the discovery of some new Hopf algebras: the Hecke-Hopf algebras which contain Hecke algebras and co-act on them at the same time. The new information resulting from this study allows for the construction of new solutions to the quantum Yang-Baxter equation. Another project in the proposal is to explore new bases in quantized enveloping algebras which possess remarkable properties such as braid group action and the compatibility with Joseph's decomposition of the quantized enveloping algebras. These new bases are expected to settle the problem of decomposing endomorphism algebras of representations and to help explicitly compute the center of the ambient algebra. New information resulting from this study will be applied to constructing new quantum and noncommutative cluster structures and to the proof of the quantum Gelfand-Kirillov conjecture for those algebras.