Symmetric quantum Weyl algebras
Annales Mathématiques Blaise Pascal, Tome 11 (2004) no. 2, pp. 187-203.

We study the symmetric powers of four algebras: $q$-oscillator algebra, $q$-Weyl algebra, $h$-Weyl algebra and $U\left({\mathrm{𝔰𝔩}}_{2}\right)$. We provide explicit formulae as well as combinatorial interpretation for the normal coordinates of products of arbitrary elements in the above algebras.

@article{AMBP_2004__11_2_187_0,
author = {D\'\i az, Rafael and Pariguan, Eddy},
title = {Symmetric quantum Weyl algebras},
journal = {Annales Math\'ematiques Blaise Pascal},
pages = {187--203},
publisher = {Annales math\'ematiques Blaise Pascal},
volume = {11},
number = {2},
year = {2004},
doi = {10.5802/ambp.192},
mrnumber = {2109607},
zbl = {02205936},
language = {en},
url = {http://www.numdam.org/item/AMBP_2004__11_2_187_0/}
}
Díaz, Rafael; Pariguan, Eddy. Symmetric quantum Weyl algebras. Annales Mathématiques Blaise Pascal, Tome 11 (2004) no. 2, pp. 187-203. doi : 10.5802/ambp.192. http://www.numdam.org/item/AMBP_2004__11_2_187_0/

[1] Alev, J.; Hodges, T.J.; Velez, J.D. Fixed rings of the Weyl algebra ${A}_{1}\left(ℂ\right)$, Journal of algebra, Volume 130 (1990), pp. 83-96 | Article | MR 1045737 | Zbl 0695.16022

[2] Kassel, Christian Quantum groups, Springer-Velarg, New York, 1995 | MR 1321145 | Zbl 0808.17003

[3] Martinec, Emil; Moore, Gregory Noncommutative Solitons on Orbifolds (2001) (hep-th/0101199)

[4] Kontsevich, M. Deformation Quantization of Poisson Manifolds I (1997) (math. q-alg/97090401)

[5] Korogodski, Leonid; Soibelman., Yan Algebras of functions on Quantum groups. Part I, Mathematical surveys and monographs, Volume 56 (1996) | Zbl 0923.17017

[6] Etingof, Pavel; Ginzburg, Victor Symplectic reflection algebras, Calogero-Moser space, and deformed Harish-Chandra homomorphism, Invent. Math, Volume 147 (2002) no. 2, pp. 243-348 | Article | MR 1881922 | Zbl 1061.16032

[7] Doubilet, Peter On the foundations of combinatorial theory. VII: Symmetric functions through the theory of distribution and occupancy, Gian-Carlo Rota on Combinatorics. Introductory papers and commentaries (1995), pp. 402-422

[8] Díaz, Rafael; Pariguan, Eddy Quantum symmetric functions (2003) (math.QA/0312494)

[9] Díaz, Rafael; Pariguan, Eddy Super, quantum and non-commutative species (2004) (Work in progress)

[10] Gopakumar, Rajesh; Minwalla, Shiraz; Strominger, Andrew Noncommutative Solitons, J. High Energy Phys. JHEP, Volume 05-020 (2000) | MR 1768736 | Zbl 0989.81612

[11] Solomon, A.I. Phys.Lett. A 196, 1994 (Number 29, Volume 126)

[12] Kac, Victor Infinite dimensional Lie algebras., Cambridge University Press, New York, 1990 | MR 1104219 | Zbl 0716.17022

[13] Kac, Victor; Cheung, Pokman Quantum Calculus, Springer-Velarg, New York, 2002 | MR 1865777 | Zbl 0986.05001