×

zbMATH — the first resource for mathematics

Combinatorial Hopf algebras and towers of algebras – dimension, quantization and functorality. (English) Zbl 1281.16036
Let \(A=\bigoplus_{n\geq 0}A_n\) be a tower of algebras as defined by N. Bergeron and H. Li [J. Algebra 321, No. 8, 2068-2084 (2009; Zbl 1185.16008)], where they show that for such a tower the Grothendieck groups of categories of finitely generated projective left \(A\)-modules, respectively of all left \(A\)-modules, form a pair of graded dual Hopf algebras. In the paper under review it is proved that in the above framework one has \(\dim(A_n)=\dim(A_1)^n\cdot n!\) for any \(n\). A quantum version of this result, involving \(q\)-twisted Hopf algebras, is also proved.

MSC:
16T05 Hopf algebras and their applications
16W50 Graded rings and modules (associative rings and algebras)
16E20 Grothendieck groups, \(K\)-theory, etc.
16T20 Ring-theoretic aspects of quantum groups
PDF BibTeX XML Cite
Full Text: DOI arXiv arXiv
References:
[1] Aguiar, M., Bergeron, N., Sottile, F.: Combinatorial Hopf algebras and generalized Dehn-Sommerville relations.Compos. Math. 142(1), 1–30 (2006) · Zbl 1092.05070 · doi:10.1112/S0010437X0500165X
[2] Aguiar, M., Sottile, F.: Structure of the Malvenuto–Reutenauer Hopf algebra of permutations. Adv. Math. 191(2), 225–275 (2005) · Zbl 1056.05139 · doi:10.1016/j.aim.2004.03.007
[3] Ariki, S.: On the decomposition numbers of the Hecke algebra of G(n, 1, m). J. Math. Kyoto Univ. 36, 789–808 (1996) · Zbl 0888.20011
[4] Bergeron, N., Hivert, F., Thibon, J.-Y.: The peak algebra and the Hecke–Clifford algebras at q = 0. J. Comb. Theory, Ser. A 107-1, 1–19 (2004) · Zbl 1107.05092 · doi:10.1016/j.jcta.2004.03.004
[5] Bergeron, N., Lam, T., Li, H.: Combinatorial Hopf algebras and towers of algebras. In: 20th Annual International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2008), Discrete Math. Theor. Comput. Sci. Proc., pp. 52–60. AJ, Assoc. Discrete Math. Theor. Comput. Sci., Nancy, France (2008)
[6] Bergeron, N., Li, H.: Algebraic structures on Grothendieck groups of a tower of algebras. J. Algebra 321, 2068–2084 (2009) · Zbl 1185.16008 · doi:10.1016/j.jalgebra.2008.12.005
[7] Brundan, J.: Modular branching rules and the Mullineux map for Hecke algebras of type A. Proc. Lond. Math. Soc. 77, 551–581 (1998) · Zbl 0904.20007 · doi:10.1112/S0024611598000562
[8] Burroughs, J.: Operations in Grothendieck rings and the symmetric group. Can. J. Math. 26, 543–550 (1974) · Zbl 0307.55016 · doi:10.4153/CJM-1974-050-3
[9] Crane, L., Frenkel, I.: Four-dimensional topological quantum field theory, Hopf categories, and the canonical bases. J. Math. Phys. 35, 5136–5154 (1994) · Zbl 0892.57014 · doi:10.1063/1.530746
[10] Curtis, C., Reiner, I.: Methods of Representation Theory, vol. I. With Applications to Finite Groups and Orders, Wiley, New York (1990) · Zbl 0698.20001
[11] Fomin, S.: Duality of graded graphs. J. Algebr. Comb. 3(4), 357–404 (1994) · Zbl 0810.05005 · doi:10.1023/A:1022412010826
[12] Geissinger, L.: Hopf algebras of symmetric functions and class functions. Combinatoire et combinatoire et représentation du groupe symétrique (Actes Table Ronde C.N.R.S., Univ. Louis-Pasteur Strasbourg, Strasbourg, 1976). In: Lecture Notes in Math., vol. 579, pp. 168–181. Springer, Berlin (1977)
[13] Hivert, F., Novelli, J.-C.,Thibon, J.-Y.: Representation theory of the 0-Ariki-Koike-Shoji algebras. Adv. Math. 205(2), 504–548 (2006) · Zbl 1105.20003 · doi:10.1016/j.aim.2005.07.016
[14] Hivert, F., Novelli, J.-C.,Thibon, J.-Y.: Commutative combinatorial Hopf algebras. J. Algebr. Comb. 28-1, 65–95 (2008) · Zbl 1181.16031 · doi:10.1007/s10801-007-0077-0
[15] Hivert, F., Nzeutchap, J.: Dual Graded Graphs in Combinatorial Hopf Algebras (in preparation). http://www.fields.utoronto.ca/\(\sim\)janvier/docs/dual_graphs_hopf_algebras.pdf
[16] Joyal, A., Street, R.: Braided tensor categories. Adv. Math. 102(1), 20–78 (1993) · Zbl 0817.18007 · doi:10.1006/aima.1993.1055
[17] Khovanov, M.: NilCoxeter algebras categorify the Weyl algebra. Commun. Algebra 29(11), 5033–5052 (2001) · Zbl 1018.16015 · doi:10.1081/AGB-100106800
[18] Krob, D., Thibon, J.-Y.: Noncommutative symmetric functions. IV. Quantum linear groups and Hecke algebras at q = 0. J. Algebr. Comb. 6-4, 339–376 (1997) · Zbl 0881.05120 · doi:10.1023/A:1008673127310
[19] Lam, T.: Quantized dual graded graphs. Electron. J. Comb. 17(1), R88 (2010) · Zbl 1230.05163
[20] Lam, T., Shimozono, M.: Dual graded graphs for Kac–Moody algebras. Algebra and Number Theory 1, 451–488 (2007) · Zbl 1200.05249 · doi:10.2140/ant.2007.1.451
[21] Lascoux, A., Leclerc, B., Thibon, J.-Y.: Hecke algebras at roots of unity and canonical bases of quantum affine algebras. Commun. Math. Phys. 181, 205–263 (1996) · Zbl 0874.17009 · doi:10.1007/BF02101678
[22] Li, H.: Algebraic Structures on Grothendieck Groups of a Tower of Algebras. Ph. D. Thesis, York University, 116 pp. (2007). http://www.math.drexel.edu/\(\sim\)huilan/thesis.pdf
[23] Li, L., Zhang, P.: Twisted Hopf algebras, Ringel–Hall algebras, and Green’s categories. J. Algebra 231, 713–743 (2000) · Zbl 0995.16031 · doi:10.1006/jabr.2000.8362
[24] Loday, J.-L.: Generalized bialgebras and triples of operads. Astérisque 320, vi+114 pp. (2008)
[25] Malvenuto, C., Reutenauer, C.: Duality between quasi-symmetric functions and the solomon descent algebra. J. Algebra 177-3, 967–982 (1995) · Zbl 0838.05100 · doi:10.1006/jabr.1995.1336
[26] Sergeev, A.N.: Tensor algebra of the identity representation as a module over the Lie superalgebras GL(n;m) and Q(n). Math. USSR Sbornik 51, 419–427 (1985) · Zbl 0573.17002 · doi:10.1070/SM1985v051n02ABEH002867
[27] Stanley, R.: Differential posets. J. Am. Math. Soc. 1, 919–961 (1988) · Zbl 0658.05006 · doi:10.1090/S0894-0347-1988-0941434-9
[28] Thibon, J.-Y., Ung, B.-C.-V.: Quantum quasi-symetric functions and Hecke algebras. J. Phys., A, Math. Gen. 29, 7337–7348 (1996) · Zbl 0962.05060 · doi:10.1088/0305-4470/29/22/027
[29] Zelevinsky, A.V.: Representations of finite classical groups. A Hopf algebra approach. In: Lecture Notes in Mathematics, vol. 869. Springer, Berlin (1981) · Zbl 0465.20009
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.