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A parametric family of subalgebras of the Weyl algebra I. Structure and automorphisms


Authors: Georgia Benkart, Samuel A. Lopes and Matthew Ondrus
Journal: Trans. Amer. Math. Soc. 367 (2015), 1993-2021
MSC (2010): Primary 16S32; Secondary 16W20
DOI: https://doi.org/10.1090/S0002-9947-2014-06144-8
Published electronically: November 18, 2014
MathSciNet review: 3286506
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Abstract: An Ore extension over a polynomial algebra $ \mathbb{F}[x]$ is either a quantum plane, a quantum Weyl algebra, or an infinite-dimensional unital associative algebra $ \mathsf {A}_h$ generated by elements $ x,y$, which satisfy $ yx-xy = h$, where $ h\in \mathbb{F}[x]$. We investigate the family of algebras $ \mathsf {A}_h$ as $ h$ ranges over all the polynomials in $ \mathbb{F}[x]$. When $ h \neq 0$, the algebras $ \mathsf {A}_h$ are subalgebras of the Weyl algebra $ \mathsf {A}_1$ and can be viewed as differential operators with polynomial coefficients. We give an exact description of the automorphisms of $ \mathsf {A}_h$ over arbitrary fields $ \mathbb{F}$ and describe the invariants in $ \mathsf {A}_h$ under the automorphisms. We determine the center, normal elements, and height one prime ideals of $ \mathsf {A}_h$, localizations and Ore sets for $ \mathsf {A}_h$, and the Lie ideal $ [\mathsf {A}_h,\mathsf {A}_h]$. We also show that $ \mathsf {A}_h$ cannot be realized as a generalized Weyl algebra over $ \mathbb{F}[x]$, except when $ h \in \mathbb{F}$. In two sequels to this work, we completely describe the irreducible modules and derivations of $ \mathsf {A}_h$ over any field.


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Additional Information

Georgia Benkart
Affiliation: Department of Mathematics, University of Wisconsin-Madison, Madison, Wisconsin 53706
Email: benkart@math.wisc.edu

Samuel A. Lopes
Affiliation: CMUP, Faculdade de Ciências, Universidade do Porto, Rua do Campo Alegre 687, 4169-007 Porto, Portugal
Email: slopes@fc.up.pt

Matthew Ondrus
Affiliation: Department of Mathematics, Weber State University, Ogden, Utah 84408
Email: mattondrus@weber.edu

DOI: https://doi.org/10.1090/S0002-9947-2014-06144-8
Received by editor(s): October 17, 2012
Received by editor(s) in revised form: April 6, 2013
Published electronically: November 18, 2014
Additional Notes: This research was funded by the European Regional Development Fund through the programme COMPETE and by the Portuguese Government through the FCT – Fundação para a Ciência e a Tecnologia under the project PEst-C/MAT/UI0144/2011.
Article copyright: © Copyright 2014 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.