Exponentiations over the quantum algebra U q (sl 2 ())
Confluentes Mathematici, Tome 5 (2013) no. 2, pp. 49-77.

We define and compare, by model-theoretical methods, some exponentiations over the quantum algebra U q (sl 2 ()). We discuss two cases, according to whether the parameter q is a root of unity. We show that the universal enveloping algebra of sl 2 () embeds in a non-principal ultraproduct of U q (sl 2 ()), where q varies over the primitive roots of unity.

Reçu le :
Révisé le :
Accepté le :
Publié le :
DOI : https://doi.org/10.5802/cml.8
Classification : 03C60,  16W35,  20G42,  81R50
Mots clés : Quantum algebra, quantum plane, exponential map, ultraproduct
     author = {Sonia L{\textquoteright}Innocente and Fran\c{c}oise Point and Carlo Toffalori},
     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://cml.centre-mersenne.org/articles/10.5802/cml.8/}
Sonia L’Innocente; Françoise Point; Carlo Toffalori. Exponentiations over the quantum algebra $U_{q}(sl_2(\mathbb{C}))$. Confluentes Mathematici, Tome 5 (2013) no. 2, pp. 49-77. doi : 10.5802/cml.8. https://cml.centre-mersenne.org/articles/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 sl 2 (). 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.