A uniform approach to the Damiani, Beck, and alternating PBW bases for the positive part of $U_q(\hat{\mathfrak{sl}}_2)$

Abstract: The $q$-deformed enveloping algebra $U_q(\hat{\mathfrak{sl}}_2)$ and its positive part $U^+_q$ are studied in both mathematics and mathematical physics. The literature contains at least three PBW bases for $U^+_q$, called the Damiani, the Beck, and the alternating PBW bases. These PBW bases are related via exponential formulas. In this talk, we will introduce an exponential generating function whose argument is a power series involving the Beck PBW basis and an integer parameter $m$. The cases $m = 2$ and $m = −1$ yield the known exponential formulas for the Damiani and alternating PBW bases, respectively. We will give present two results on the generating function for an arbitrary integer m. The first result gives a factorization of the generating function. In the second result, we express the coefficients of the generating function in closed form.

mathematical physicsdynamical systemsquantum algebrarepresentation theorysymplectic geometry

Audience: general audience

BIMSA Integrable Systems Seminar

Series comments: The aim is to bring together experts in integrable systems and related areas of theoretical and mathematical physics and mathematics. There will be research presentations and overview talks.

Audience: Graduate students and researchers interested in integrable systems and related mathematical structures, such as symplectic and Poisson geometry and representation theory.

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