Lagrange's Theorem for Hopf Monoids in Species

Aguiar, Marcelo ; Lauve, Aaron

Canadian journal of mathematics, Tome 65 (2013) no. 2, pp. 241-265

Résumé

Following Radford's proof of Lagrange's theorem for pointed Hopf algebras, we prove Lagrange‘s theorem for Hopf monoids in the category of connected species. As a corollary, we obtain necessary conditions for a given subspecies $\mathbf{k}$ of a Hopf monoid $\mathbf{h}$ to be a Hopf submonoid: the quotient of any one of the generating series of $\mathbf{h}$ by the corresponding generating series of $\mathbf{k}$ must have nonnegative coefficients. Other corollaries include a necessary condition for a sequence of nonnegative integers to be the dimension sequence of a Hopf monoid in the form of certain polynomial inequalities and of a set-theoretic Hopf monoid in the form of certain linear inequalities. The latter express that the binomial transform of the sequence must be nonnegative.

Détail

DOI : 10.4153/CJM-2011-098-9

Mots-clés : 05A15, 05A20, 05E99, 16T05, 16T30, 18D10, 18D35, Hopf monoids, species, graded Hopf algebras, Lagrange's theorem, generating series, Poincaré–Birkhoff–Witt theorem, Hopf kernel, Lie kernel, primitive element, partition, composition, linear order, cyclic order, derangement

Bibliographie

[1] [1] Abe, E., Hopf algebras. Translated from the Japanese by Hisae Kinoshita and Hiroko Tanaka, Cambridge Tracts in Mathematics, 74, Cambridge University Press, Cambridge-New York, 1980. Google Scholar

[2] [2] Aguiar, M. and Mahajan, S., Monoidal functors, species and Hopf algebras. CRM Monograph Series, 29, American Mathematical Society, Providence, RI, 2010. Google Scholar

[3] [3] Aguiar, M. and Sottile, F., Cocommutative Hopf algebras of permutations and trees. J. Algebraic Combin. 22(2005), 451–470. Google Scholar | DOI

[4] [4] Bergeron, F., Labelle, G., and Leroux, P., Combinatorial species and tree-like structures. Translated from the 1994 French original by Margaret Readdy, with a foreword by Gian-Carlo Rota, Encyclopedia of Mathematics and its Applications, 67, Cambridge University Press, Cambridge, 1998. Google Scholar

[5] [5] Blattner, R. J., Cohen, M., and Montgomery, S., Crossed products and inner actions of Hopf algebras. Trans. Amer. Math. Soc. 298(1986), 671–711. Google Scholar | DOI

[6] [6] Dekker, J. C. E., Myhill's theory of combinatorial functions. Modern Logic 1(1990), no. 1, 3–21. Google Scholar

[7] [7] Garsia, A. M. and Wallach, N., Qsym over Sym is free. J. Combin. Theory Ser. A 104(2003), no. 2, 217–263. Google Scholar | DOI

[8] [8] Jacobson, N., Lie algebras. Republication of the 1962 original, Dover Publications Inc. , New York,1979. Google Scholar

[9] [9] Joyal, A., Une théorie combinatoire des séries formelles. Adv. in Math. 42(1981), no. 1, 1–82. Google Scholar | DOI

[10] [10] Kaplansky, I., Bialgebras. Lecture Notes in Mathematics, Department of Mathematics, University of Chicago, Chicago, Ill., 1975. Google Scholar

[11] [11] Larson, R. G. and Sweedler, M. E., An associative orthogonal bilinear form for Hopf algebras. Amer. J. Math. 91(1969), 75–94. Google Scholar | DOI

[12] [12] Lauve, A. and Mason, S. K., Qsym over sym has a stable basis. J. Combin. Theory Ser. A 118(2011), no. 5, 1661–1673. Google Scholar | DOI

[13] [13] Lothaire, M.. Combinatorics on words.With a foreword by Roger Lyndon and a preface by Dominique Perrin. Corrected reprint of the 1983 original, with a new preface by Perrin, Cambridge Mathematical Library, Cambridge University Press, Cambridge, 1997. Google Scholar

[14] [14] Lyubashenko, V., Modular transformations for tensor categories. J. Pure Appl. Algebra 98(1995), no. 3, 279–327. Google Scholar | DOI

[15] [15] Macdonald., I. G. Symmetric functions and Hall polynomials Second ed., Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1995. Google Scholar

[16] [16] Masuoka, A., Freeness of Hopf algebras over coideal subalgebras. Comm. Algebra 20(1992), no. 5, 1353–1373. Google Scholar | DOI

[17] [17] Menni, M.. Algebraic categories whose projectives are explicitly free. Theory Appl. Categ. 22(2009), no. 20, 509–541. Google Scholar

[18] [18] Montgomery, S., Hopf algebras and their actions on rings. CBMS Regional Conference Series in Mathematics, 82, Published for the Conference Board of the Mathematical Sciences,Washington, DC; by the American Mathematical Society, Providence, RI, 1993. Google Scholar

[19] [19] Myhill, J., Recursive equivalence types and combinatorial functions. Bull. Amer. Math. Soc. 64(1958), 373–376. Google Scholar | DOI

[20] [20] Nichols, W. D. and Zoeller, M. B., A Hopf algebra freeness theorem. Amer. J. Math. 111(1989), no. 2, 381–385. Google Scholar | DOI

[21] [21] Oberst, U. and Schneider, H.-J., Untergruppen formeller Gruppen von endlichem Index. J. Algebra 31(1974), 10–44. Google Scholar | DOI

[22] [22] Radford, D. E., Pointed Hopf algebras are free over Hopf subalgebras. J. Algebra 45(1977), no. 2, 266–273. Google Scholar | DOI

[23] [23] Radford, D. E., A natural ring basis for the shuffle algebra and an application to group schemes. J. Algebra 58(1979), no. 2, 432–454. Google Scholar | DOI

[24] [24] Reutenauer, C., Free Lie algebras. London Mathematical Society Monographs. New Series, 7, Oxford Science Publications, The Clarendon Press, Oxford University Press, New York, 1993. Google Scholar

[25] [25] Reutenauer, C., Free Lie algebras. In: Handbook of algebra, 3, North-Holland, Amsterdam, 2003, pp. 887–903. Google Scholar

[26] [26] Rosales, J. C. and Garcίa-Sánchez, P. A., Numerical semigroups. Developments in Mathematics, 20, Springer, New York, 2009. Google Scholar

[27] [27] Schneider, H.-J., Principal homogeneous spaces for arbitrary Hopf algebras. Israel J. Math. 72(1990), no. 1–2, 167–195. Google Scholar | DOI

[28] [28] Schneider, H.-J., Normal basis and transitivity of crossed products for Hopf algebras. J. Algebra 152(1992), no. 2, 289–312. Google Scholar | DOI

[29] [29] Sloane, N. J. A., The on-line encyclopedia of integer sequences. http://www.research.att.com/_njas/sequences, OEIS. Google Scholar

[30] [30] Sommerhäuser, Y., On Kaplansky's conjectures. In: Interactions between ring theory and representations of algebras (Murcia), Lecture Notes in Pure and Appl. Math., 210, Dekker, New York, 2000, pp. 393–412. Google Scholar

[31] [31] Stanley, R. P., Enumerative combinatorics. 2. Cambridge Studies in Advanced Mathematics, 62, Cambridge University Press, Cambridge, 1999. Google Scholar

[32] [32] M. Takeuchi, , Relative Hopf modules—equivalences and freeness criteria. J. Algebra 60(1979), no. 2, 452–471. Google Scholar | DOI

[33] [33] M. Takeuchi, , Finite Hopf algebras in braided tensor categories. J. Pure Appl. Algebra 138, no. 1, 59–82. Google Scholar | DOI

[34] [34] Wang, D. G., Zhang, J. J., and Zhuang, G.. Hopf algebras of GK-dimension two with vanishing Ext-group. arxiv:1105.0033v1 Google Scholar