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Affine Actions of Uq(sl(2)) on Polynomial Rings

Published online by Cambridge University Press:  20 November 2018

Jeffrey Bergen
Jeffrey Bergen*
Affiliation:
Department of Mathematics, DePaul University, 2320 N. Kenmore Avenue, Chicago, Illinois 60614, USA. e-mail: [email protected]
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Abstract

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We classify the affine actions of ${{U}_{q}}\left( sl\left( 2 \right) \right)$ on commutative polynomial rings in $m\,\ge \,1$ variables. We show that, up to scalar multiplication, there are two possible actions. In addition, for each action, the subring of invariants is a polynomial ring in either $m$ or $m\,-\,1$ variables, depending upon whether $q$ is or is not a root of 1.

Keywords

16T2017B3720G42skew derivationquantum groupinvariants
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 58 , Issue 2 , 01 June 2015 , pp. 233 - 240
Copyright
Copyright © Canadian Mathematical Society 2015

References

[1] Brown, K. A. and Goodearl, K. R., Lectures on algebraic quantum groups. Advanced Courses in Mathematics. CRM Barcelona, Birkhâuser Verlag, Basel, 2002.Google Scholar
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[3] Hu, N., Montgomery, S. and Smith, S. P., Skew derivations and Uq(sl(2)). Israel J. Math. 72 (1990), no. 1–2, 158–166. http://dx.doi.org/10.1007/BF02764618 Google Scholar