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arXiv · 1101.4912

Quantum affine algebras, canonical bases and $q$-deformation of arithmetical functions

Henry H. Kim·Kyu-Hwan Lee
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Abstract

In this paper, we obtain affine analogues of Gindikin-Karpelevich formula and Casselman-Shalika formula as sums over Kashiwara-Lusztig's canonical bases. Suggested by these formulas, we define natural $q$-deformation of arithmetical functions such as (multi-)partition function and Ramanujan $τ$-function, and prove various identities among them. In some examples, we recover classical identities by taking limits. We also consider $q$-deformation of Kostant's function and study certain $q$-polynomials whose special values are weight multiplicities.

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BibTeXRIS

Henry H. Kim, Kyu-Hwan Lee. 2012-03-23. Quantum affine algebras, canonical bases and $q$-deformation of arithmetical functions. https://arxiv.org/abs/1101.4912

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