Abstract
Bergeron and Li have introduced a set of axioms which guarantee that the Grothendieck groups of a tower of algebras \(\bigoplus_{n\ge0}A_n\) can be a pair of graded dual Hopf algebras. Hivert and Nzeutchap, and independently Lam and Shimozono constructed dual graded graphs from primitive elements in Hopf algebras. In this paper we apply the composition of these constructions to towers of algebras. We show that if a tower \(\bigoplus_{n\ge0}A_n\) gives rise to a pair of graded dual Hopf algebras, then \(\dim(A_n)=r^nn!\) where \(r = \dim(A_1)\). In the case of r = 1 we give a conjectural classification. We then investigate a quantum version of the main theorem. We conclude with some open problems and a categorification of these constructions.
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N. Bergeron is supported in part by CRC and NSERC. T. Lam is partially supported by NSF grants DMS-0600677 and DMS-0652641. H. Li is supported in part by CRC, NSERC and NSF grant DMS-0652641.
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Bergeron, N., Lam, T. & Li, H. Combinatorial Hopf Algebras and Towers of Algebras—Dimension, Quantization and Functorality. Algebr Represent Theor 15, 675–696 (2012). https://doi.org/10.1007/s10468-010-9258-y
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DOI: https://doi.org/10.1007/s10468-010-9258-y