no. 5
Character groups of Hopf algebras as infinite-dimensional Lie groups
[Groupes de caractères des algèbres de Hopf vus comme groupes de Lie de dimension infinie]
Bogfjellmo, Geir1 ; Dahmen, Rafael2 ; Schmeding, Alexander1
1 NTNU Trondheim Alfred Getz’ vei 1 7491 Trondheim (Norway)
2 Fachbereich Mathematik, TU Darmstadt Schloßgartenstr. 7 64289 Darmstadt (Germany)
Annales de l'Institut Fourier, Tome 66 (2016) no. 5, pp. 2101-2155.

Dans cet article, nous étudions les groupes de caractères des algèbres de Hopf du point de vue de la théorie de Lie de dimension infinie. Pour une algèbre de Hopf connexe et graduée, nous munissons le groupe de caractères d’une structure de groupe de Lie de dimension infinie, à valeurs dans une algèbre localement convexe. Cette structure permet de voir le groupe de caractères comme un groupe de Lie de Baker–Campbell–Hausdorff, qui est régulier au sens de Milnor. De plus, nous montrons que certains sous-groupes associés aux idéaux de Hopf sont alors des sous-groupes de Lie du groupe de caractères.

Si l’algèbre de Hopf n’est pas graduée, son groupe de caractères ne sera pas un groupe de Lie, en général. Cependant, nous montrons que pour une algèbre de Hopf quelconque, le groupe de caractères à valeurs dans une algèbre faiblement complète est un groupe pro-Lie au sens de Hofmann et Morris.

In this article character groups of Hopf algebras are studied from the perspective of infinite-dimensional Lie theory. For a graded and connected Hopf algebra we obtain an infinite-dimensional Lie group structure on the character group with values in a locally convex algebra. This structure turns the character group into a Baker–Campbell–Hausdorff–Lie group which is regular in the sense of Milnor. Furthermore, we show that certain subgroups associated to Hopf ideals become closed Lie subgroups of the character group.

If the Hopf algebra is not graded, its character group will in general not be a Lie group. However, we show that for any Hopf algebra the character group with values in a weakly complete algebra is a pro-Lie group in the sense of Hofmann and Morris.

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DOI : 10.5802/aif.3059
Classification : 22E65, 16T05, 43A40, 58B25, 46H30, 22A05
Keywords: real analytic, infinite-dimensional Lie group, Hopf algebra, continuous inverse algebra, Butcher group, weakly complete space, pro-Lie group
Mot clés : réel analytique, groupe de Lie de dimension infinie, algèbres de Hopf, algèbre avec inversion continue, espace faiblement complet, groupe pro-Lie

Bogfjellmo, Geir 1 ; Dahmen, Rafael 2 ; Schmeding, Alexander 1

1 NTNU Trondheim Alfred Getz’ vei 1 7491 Trondheim (Norway)
2 Fachbereich Mathematik, TU Darmstadt Schloßgartenstr. 7 64289 Darmstadt (Germany) 
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Bogfjellmo, Geir; Dahmen, Rafael; Schmeding, Alexander. Character groups of Hopf algebras as infinite-dimensional Lie groups. Annales de l'Institut Fourier, Tome 66 (2016) no. 5, pp. 2101-2155. doi : 10.5802/aif.3059. https://aif.centre-mersenne.org/articles/10.5802/aif.3059/

[1] Bertram, W.; Glöckner, H.; Neeb, K.-H. Differential calculus over general base fields and rings, Expo. Math., Volume 22 (2004) no. 3, pp. 213-282 | DOI

[2] Bochnak, Jacek; Siciak, Józef Polynomials and multilinear mappings in topological vector spaces, Studia Math., Volume 39 (1971), pp. 59-76

[3] Bogfjellmo, Geir; Schmeding, Alexander The Lie Group Structure of the Butcher Group, Foundations of Computational Mathematics (2015), pp. 1-33 (http://dx.doi.org/10.1007/s10208-015-9285-5) | DOI

[4] Brouder, Ch. Trees, renormalization and differential equations, BIT, Volume 44 (2004) no. 3, pp. 425-438 | DOI

[5] Cartier, Pierre A primer of Hopf algebras, Frontiers in number theory, physics, and geometry. II, Springer, Berlin, 2007, pp. 537-615 | DOI

[6] Chartier, Philippe; Hairer, Ernst; Vilmart, Gilles Algebraic structures of B-series, Found. Comput. Math., Volume 10 (2010) no. 4, pp. 407-427 | DOI

[7] Connes, Alain; Kreimer, Dirk Hopf algebras, renormalization and noncommutative geometry, Comm. Math. Phys., Volume 199 (1998) no. 1, pp. 203-242 | DOI

[8] Connes, Alain; Kreimer, Dirk Renormalization in quantum field theory and the Riemann-Hilbert problem. I. The Hopf algebra structure of graphs and the main theorem, Comm. Math. Phys., Volume 210 (2000) no. 1, pp. 249-273 | DOI

[9] Connes, Alain; Kreimer, Dirk Renormalization in quantum field theory and the Riemann-Hilbert problem. II. The β-function, diffeomorphisms and the renormalization group, Comm. Math. Phys., Volume 216 (2001) no. 1, pp. 215-241 | DOI

[10] Connes, Alain; Marcolli, Matilde Noncommutative geometry, quantum fields and motives, American Mathematical Society Colloquium Publications, 55, American Mathematical Society, Providence, RI; Hindustan Book Agency, New Delhi, 2008, xxii+785 pages

[11] Glöckner, Helge Algebras whose groups of units are Lie groups, Studia Math., Volume 153 (2002) no. 2, pp. 147-177 | DOI

[12] Glöckner, Helge Infinite-dimensional Lie groups without completeness restrictions, Geometry and analysis on finite- and infinite-dimensional Lie groups (Będlewo, 2000) (Banach Center Publ.), Volume 55, Polish Acad. Sci., Warsaw, 2002, pp. 43-59

[13] Glöckner, Helge Instructive examples of smooth, complex differentiable and complex analytic mappings into locally convex spaces, J. Math. Kyoto Univ., Volume 47 (2007) no. 3, pp. 631-642

[14] Glöckner, Helge Simplified proofs for the pro-Lie group theorem and the one-parameter subgroup lifting lemma, J. Lie Theory, Volume 17 (2007) no. 4, pp. 899-902

[15] Glöckner, Helge Regularity properties of infinite-dimensional Lie groups, and semiregularity (2015) (http://arxiv.org/abs/1208.0715v3)

[16] Glöckner, Helge; Neeb, Karl-Hermann When unit groups of continuous inverse algebras are regular Lie groups, Studia Math., Volume 211 (2012) no. 2, pp. 95-109 | DOI

[17] Hofmann, K. H.; Neeb, K.-H. Pro-Lie groups which are infinite-dimensional Lie groups, Math. Proc. Cambridge Philos. Society, Volume 146 (2009) no. 2, pp. 351-378 | DOI

[18] Hofmann, Karl H.; Morris, Sidney A. The Lie theory of connected pro-Lie groups, EMS Tracts in Mathematics, 2, EMS, Zürich, 2007, xvi+678 pages | DOI

[19] Hofmann, Karl H.; Morris, Sidney A. The structure of compact groups, De Gruyter Studies in Mathematics, 25, De Gruyter, Berlin, 2013, xxii+924 pages | DOI

[20] Kassel, Christian Quantum groups, Graduate Texts in Mathematics, 155, Springer-Verlag, New York, 1995, xii+531 pages | DOI

[21] Keller, Hans Heinrich Differential calculus in locally convex spaces, Lecture Notes in Mathematics, Vol. 417, Springer-Verlag, Berlin-New York, 1974, iii+143 pages

[22] Kriegl, Andreas; Michor, Peter W. The convenient setting of global analysis, Mathematical Surveys and Monographs, 53, American Mathematical Society, Providence, RI, 1997, x+618 pages

[23] Larson, Richard G. Cocommutative Hopf algebras, Canad. J. Math., Volume 19 (1967), pp. 350-360 | DOI

[24] Loday, Jean-Louis; Ronco, María Combinatorial Hopf algebras, Quanta of maths (Clay Math. Proc.), Volume 11, Amer. Math. Soc., Providence, RI, 2010, pp. 347-383

[25] Majid, Shahn Foundations of quantum group theory, Cambridge University Press, Cambridge, 1995, x+607 pages | DOI

[26] Manchon, Dominique Hopf algebras, from basics to applications to renormalization (2006) (http://arxiv.org/pdf/math/0408405v2)

[27] Michaelis, Walter Coassociative coalgebras, Handbook of algebra, Vol. 3, North-Holland, Amsterdam, 2003, pp. 587-788 | DOI

[28] Milnor, John W. Remarks on infinite-dimensional Lie groups, Relativity, groups and topology, II (Les Houches, 1983), North-Holland, Amsterdam, 1984, pp. 1007-1057

[29] Milnor, John W.; Moore, John C. On the structure of Hopf algebras, Ann. of Math. (2), Volume 81 (1965), pp. 211-264 | DOI

[30] Neeb, Karl-Hermann Towards a Lie theory of locally convex groups, Jpn. J. Math., Volume 1 (2006) no. 2, pp. 291-468 | DOI

[31] van Suijlekom, Walter D. Renormalization of gauge fields: a Hopf algebra approach, Comm. Math. Phys., Volume 276 (2007) no. 3, pp. 773-798 | DOI

[32] van Suijlekom, Walter D. The structure of renormalization Hopf algebras for gauge theories. I. Representing Feynman graphs on BV-algebras, Comm. Math. Phys., Volume 290 (2009) no. 1, pp. 291-319 | DOI

[33] Sweedler, Moss E. Hopf algebras, Mathematics Lecture Note Series, W. A. Benjamin, Inc., New York, 1969, vii+336 pages

[34] Waterhouse, William C. Introduction to affine group schemes, Graduate Texts in Mathematics, 66, Springer-Verlag, New York-Berlin, 1979, xi+164 pages

[35] Yamabe, Hidehiko On the conjecture of Iwasawa and Gleason, Ann. of Math. (2), Volume 58 (1953), pp. 48-54 | DOI

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