Tannakian Categories With SemigroupActions

Ovchinnikov, Alexey ; Wibmer, Michael

Canadian journal of mathematics, Tome 69 (2017) no. 3, pp. 687-720

Résumé

A theorem of Ostrowski implies that $\log \left( x \right),\,\log \left( x+1 \right),\,.\,.\,.$ are algebraically independent over $\mathbb{C}\left( x \right)$ . More generally, for a linear differential or difference equation, it is an important problem to find all algebraic dependencies among a non-zero solution $y$ and particular transformations of $y$ , such as derivatives of $y$ with respect to parameters, shifts of the arguments, rescaling, etc. In this paper, we develop a theory of Tannakian categories with semigroup actions, which will be used to attack such questions in full generality, as each linear differential equation gives rise to a Tannakian category. Deligne studied actions of braid groups on categories and obtained a finite collection of axioms that characterizes such actions to apply them to various geometric constructions. In this paper, we find a finite set of axioms that characterizes actions of semigroups that are finite free products of semigroups of the form ${{\mathbb{N}}^{n}}\,\times \,\mathbb{Z}/{{n}_{1}}\mathbb{Z}\,\times \,.\,.\,.\,\mathbb{Z}/{{n}_{r}}\mathbb{Z}$ on Tannakian categories. This is the class of semigroups that appear in many applications.

Détail

DOI : 10.4153/CJM-2016-011-0

Mots-clés : 18D10, 12H10, 20G05, 33C05, 33C80, 34K06, semigroup actions on categories, Tannakian categories, difference algebraic groups, differential and difference equations with parameters

Bibliographie

[1] [1] Baumslag, G. and Solitar, D., Some two-generator one relator non-Hopfian groups. Bull. Amer. Math. Soc. 68(1962), 199–201. Google Scholar | DOI

[2] [2] Cohn, R. M., Difference algebra. Interscience Publishers, New York, 1965. Google Scholar

[3] [3] Deligne, P., Catégories tannakiennes. In: The Grothendieck Festschrift. II. Progr. Math. 87, Birkhäuser Boston, Boston, MA, 1990, pp. 111–195. http://dx.doi.Org/10.1007/978-0-81 76-4575-5_3 Google Scholar

[4] [4] Deligne, P., Action du groupe des tresses sur une catégorie. Invent. Math. 128(1997), 159–175. Google Scholar | DOI

[5] [5] Deligne, P. and Milne, J., Tannakian categories. In: Hodge cycles, motives, and Shimura varieties. Lecture Notes in Mathematics, 900. Springer-Verlag, Berlin, 1982, pp. 101–228. http://www.jmilne.org/math/xnotes/tc.pdf. Google Scholar

[6] [6] Di Vizio, L, Hardouin, C. , and Wibmer, M., Difference Galois theory of linear differential equations. Adv. Math. 260(2014), 1–58. Google Scholar | DOI

[7] [7] Di Vizio, L, Difference algebraic relations among solutions of linear differential equations. J. Inst. Math. Jussieu 2016. http://dx.doi.Org/10.101 7/S147474801 5000080 Google Scholar

[8] [8] Drinfeld, V., Gelaki, S., Nikshych, D., and Ostrik, V., On braided fusion categories. I. Selecta Math. (N.S.) 16(2010), no. 1,1–119. Google Scholar | DOI

[9] [9] Frenkel, E., Langlands correspondence for loop groups. Cambridge Studies in Advanced Mathematics, 103. Cambridge University Press, Cambridge, 2007. http://math.berkeley.edu/-frenkel/loop.pdf. Google Scholar

[10] [10] Gillet, H., Gorchinskiy, S., andOvchinnikov, A., Parameterized Picard-Vessiot extensions and Atiyah extensions. Adv. Math. 238(2013), 322–411. http://dx.doi.Org/10.1016/j.aim.2013.02.006 Google Scholar

[11] [11] Kamensky, M., Tannakian formalism over fields with operators. Internat. Math. Res. Notices 24(2013), 5571–5622. http://dx.doi.Org/10.1093/imrn/rns190 Google Scholar

[12] [12] Kamensky, M., Model theory and the Tannakian formalism. Trans. Amer. Math. Soc. 367(2015)1095–1120. Google Scholar | DOI

[13] [13] Koornwinder, T. H., Group theoretic interpretations of Askey's scheme of hypergeometric orthogonal polynomials. In: Orthogonal polynomials and their applications. Lecture Notes in Mathematics, 1329. Springer, 1988, pp. 46–72. http://dx.doi.Org/10.1 OO7/BFbOO83353 Google Scholar

[14] [14] Levin, A., Difference algebra. Algebra and Applications, 8. Springer, New York, 2008. Google Scholar | DOI

[15] [15] Mac Lane, S., Categories for the working mathematician. Springer, New York, 1978. Google Scholar | DOI

[16] [16] Minchenko, A. and Ovchinnikov, A., Zariski closures of reductive linear differential algebraic groups. Adv. Math. 227(2011), no. 3, 1195–1224. http://dx.doi.Org/10.1016/j.aim.2O11.03.002 Google Scholar

[17] [17] Minchenko, A., Ovchinnikov, A., and Singer, M. F., Unipotent differential algebraic groups as parameterized differential Galois groups. J. Inst. Math. Jussieu 13(2014), no. 4, 671–700. http://dx.doi.Org/10.1017/S1474748013000200 Google Scholar

[18] [18] Ostrowski, A., Sur les relations algébriques entre les intégrales indéfinies. Acta Math. 78(1946), no. 1,315–318. Google Scholar | DOI

[19] [19] Ovchinnikov, A., Tannakian approach to linear differential algebraic groups. Transform. Groups 13(2008), no. 2, 413–446. Google Scholar | DOI

[20] [20] Ovchinnikov, A., Tannakian categories, linear differential algebraic groups, and parametrized linear differential equations. Transform. Groups 14(2009), no. 1,195–223. Google Scholar | DOI

[21] [21] Ovchinnikov, A., Differential Tannakian categories. J. Algebra 321(2009), no. 10, 3043–3062. http://dx.doi.Org/10.1016/j.jalgebra.2OO9.O2.008 Google Scholar

[22] [22] Ovchinnikov, A., Difference integrability conditions for parameterized linear difference and differential equations. Adv. Appl. Math. 53(2014), 61–71. http://dx.doi.Org/10.1016/j.aam.2013.09.007 Google Scholar

[23] [23] Ovchinnikov, A. and Wibmer, M., a-Galois theory of linear difference equations. Internat. Math. Res. Notices 2015 (12): 3962–4018, 2015. http://dx.doi.Org/10.1093/imrn/rnu060 Google Scholar

[24] [24] Saavedra Rivano, N., Catégories Tannakiennes. Lecture Notes in Mathematics, 265. Springer-Verlag, Berlin, 1972. http://dx.doi.Org/10.1OO7/BFbOO59108 Google Scholar

[25] [25] van der Put, M. and Singer, M. F., Galois theory of linear differential equations. Fundamental Principles of Mathematical Sciences, 328. Springer-Verlag, Berlin, 2003. Google Scholar | DOI

[26] [26] Vilenkin, N. J.and Klimyk, A. U., Representation of Lie groups and special functions. Simplest Lie Groups, Special Functions and Integral Transforms, vol. 1. Springer, Netherlands, 1991. Google Scholar | DOI

[27] [27] Waterhouse, W. C., Introduction to affine group schemes. Graduate Texts in Mathematics, 66. Springer-Verlag, New York, 1979. Google Scholar | DOI

[28] [28] Wibmer, M., Existence of d-parameterized Picard-Vessiot extensions over fields with algebraically closed constants. J. Algebra 361(2012), 163–171. http://dx.doi.Org/10.1016/j.jalgebra.2012.03.035 Google Scholar