A parametric family of subalgebras of the Weyl algebra I. Structure and automorphisms
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- by Georgia Benkart, Samuel A. Lopes and Matthew Ondrus PDF
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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.References
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Additional Information
- Georgia Benkart
- Affiliation: Department of Mathematics, University of Wisconsin-Madison, Madison, Wisconsin 53706
- MR Author ID: 34650
- 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
- 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.
- © Copyright 2014
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - 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
- MathSciNet review: 3286506