The primitives of the Hopf algebra of noncommutative symmetric functions

2004/10/16 by Michiel Hazewinkel, Hazewinkel, Michiel · 1 citation
Mathematics · #05E05 #16W30 #Advanced Combinatorial Mathematics #Advanced Topics in Algebra #Algebraic structures and combinatorial models #Combinatorics (math.CO) #FOS: Mathematics #Quantum Algebra (math.QA) #math.CO #math.QA #msc:05E05 #msc:16W30

paper · pdf · doi:10.48550/arxiv.math/0410365

26 pages. The primitives are described over the integers (not over a characteristic zero field, which is much easier)

arxiv created 2004/10/16 · openalex publication_date 2004/10/16 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let NSymm be the Hopf algebra of noncommutative symmetric functions over the integers. In this paper a description is given of its Lie algebra of primitives over the integers, Prim(NSymm), in terms of recursion formulas. For each of the primitives of a basis of Prim(NSymm), indexed by Lyndon words, there is a recursively given divided power series over it. This gives another proof of the theorem that the algebra of quasi-symmetric functions is free over the integers.

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