We define and compare, by model-theoretical methods, some exponentiations over the quantum algebra . We discuss two cases, according to whether the parameter is a root of unity. We show that the universal enveloping algebra of embeds in a non-principal ultraproduct of , where varies over the primitive roots of unity.
Revised:
Accepted:
Published online:
Keywords: Quantum algebra, quantum plane, exponential map, ultraproduct
L’Innocente, Sonia 1; Point, Françoise 2; Toffalori, Carlo 3
@article{CML_2013__5_2_49_0,
author = {L{\textquoteright}Innocente, Sonia and Point, Fran\c{c}oise and Toffalori, Carlo},
title = {Exponentiations over the quantum algebra $U_{q}(sl_2(\mathbb{C}))$},
journal = {Confluentes Mathematici},
pages = {49--77},
publisher = {Institut Camille Jordan},
volume = {5},
number = {2},
year = {2013},
doi = {10.5802/cml.8},
language = {en},
url = {https://www.numdam.org/articles/10.5802/cml.8/}
}
TY - JOUR
AU - L’Innocente, Sonia
AU - Point, Françoise
AU - Toffalori, Carlo
TI - Exponentiations over the quantum algebra $U_{q}(sl_2(\mathbb{C}))$
JO - Confluentes Mathematici
PY - 2013
SP - 49
EP - 77
VL - 5
IS - 2
PB - Institut Camille Jordan
UR - https://www.numdam.org/articles/10.5802/cml.8/
DO - 10.5802/cml.8
LA - en
ID - CML_2013__5_2_49_0
ER -
%0 Journal Article
%A L’Innocente, Sonia
%A Point, Françoise
%A Toffalori, Carlo
%T Exponentiations over the quantum algebra $U_{q}(sl_2(\mathbb{C}))$
%J Confluentes Mathematici
%D 2013
%P 49-77
%V 5
%N 2
%I Institut Camille Jordan
%U https://www.numdam.org/articles/10.5802/cml.8/
%R 10.5802/cml.8
%G en
%F CML_2013__5_2_49_0
L’Innocente, Sonia; Point, Françoise; Toffalori, Carlo. Exponentiations over the quantum algebra $U_{q}(sl_2(\mathbb{C}))$. Confluentes Mathematici, Volume 5 (2013) no. 2, pp. 49-77. doi: 10.5802/cml.8
[1] C.C. Chang and H.J. Keisler. Model theory. North-Holland, 1973.
[2] K.R. Goodearl and R.B.Jr. Warfield. An Introduction to Noncommutative Rings. London Math. Soc. Student Texts 16, Cambridge University Press, 1989.
[3] I. Herzog and S. L’Innocente. The nonstandard quantum plane. Ann. Pure Applied Logic 156:78–85, 2008.
[4] E. Hrushovski and B. Zilber. Zariski geometries. J. Amer. Math. Soc. 9:1–56, 1996.
[5] N. Jacobson. Structure of Rings. Colloquium Publications 37, Amer. Math. Soc., 1964.
[6] J. Jantzen. Lectures on Quantum groups. Graduate Studies in Mathematics 9, Amer. Math. Soc., 1996.
[7] C. Kassel, Quantum groups. Graduate Texts in Mathematics 155, Springer, 1995.
[8] A. Klimyk and K. Schmüdgen. Quantum groups and their representations. Texts and Monographs in Physics, Springer, 1997.
[9] S. L’Innocente, A. Macintyre and F. Point. Exponentiations over the universal enveloping algebra of . Ann. Pure Applied Logic 161:1565–1580, 2010.
[10] A. Macintyre. Model theory of exponentials on Lie algebras. Math. Structures Comput. Sci. 18:189–204, 2008.
[11] M. Prest. Model theory and modules. London Math. Soc. Lecture Note Series 130, Cambridge University Press, 1987.
[12] W. Rossmann. Lie groups: An introduction through linear groups. Oxford University Press, 2002.
[13] B. Zilber. A class of quantum Zariski geometries. In Model Theory with applications to algebra and analysis, I (Z. Chatzidakis, H.D. Macpherson, A. Pillay, A.J. Wilkie editors), pages 293–326. London Math. Soc. Lecture Note Series 349, Cambridge University Press, 2008.
Cited by Sources:
