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Some q-exponential Formulas for Finite-Dimensional \(\square _{q}\)-Modules

  • Yang YangEmail author
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Abstract

We consider the algebra \(\square _{q}\) which is a mild generalization of the quantum algebra \(U_{q}(\frak {sl}_{2})\). The algebra \(\square _{q}\) is defined by generators and relations. The generators are \(\{x_{i}\}_{i\in \mathbb {Z}_{4}}\), where \(\mathbb {Z}_{4}\) is the cyclic group of order 4. For \(i\in \mathbb {Z}_{4}\) the generators xi,xi+ 1 satisfy a q-Weyl relation, and xi,xi+ 2 satisfy a cubic q-Serre relation. For \(i\in \mathbb {Z}_{4}\) we show that the action of xi is invertible on every nonzero finite-dimensional \(\square _{q}\)-module. We view \(x_{i}^{-1}\) as an operator that acts on nonzero finite-dimensional \(\square _{q}\)-modules. For \(i\in \mathbb {Z}_{4}\), define \(\mathfrak {n}_{i,i + 1}=q(1-x_{i}x_{i + 1})/(q-q^{-1})\). We show that the action of \(\mathfrak {n}_{i,i + 1}\) is nilpotent on every nonzero finite-dimensional \(\square _{q}\)-module. We view the q-exponential \(\text {{exp}}_{q}(\mathfrak {n}_{i,i + 1})\) as an operator that acts on nonzero finite-dimensional \(\square _{q}\)-modules. In our main results, for \(i,j\in \mathbb {Z}_{4}\) we express each of \(\text {{exp}}_{q}(\mathfrak {n}_{i,i + 1})x_{j}\text {{exp}}_{q}(\mathfrak {n}_{i,i + 1})^{-1}\) and \(\text {{exp}}_{q}(\mathfrak {n}_{i,i + 1})^{-1}x_{j}\text {{exp}}_{q}(\mathfrak {n}_{i,i + 1})\) as a polynomial in \(\{x_{k}^{\pm 1}\}_{k\in \mathbb {Z}_{4}}\).

Keywords

q-exponential function Quantum algebra Equitable presentation 

Mathematics Subject Classification (2010)

Primary: 33D80 Secondary: 17B37 

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Notes

Acknowledgments

This paper was written while the author was a graduate student at the University of Wisconsin-Madison. The author would like to thank his advisor, Paul Terwilliger, for offering many valuable ideas and suggestions.

As part of computational evidence, the open software SageMath (see [20]) was used to verify our main results Theorems 8.1, 8.2 and Theorems 9.3–9.6 on low dimensional irreducible \(\square _{q}\)-modules.

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© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of WisconsinMadisonUSA

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