Noncommutative symmetric function

<h2 id="definition">Definition</h2>
The underlying algebra of the Hopf algebra of noncommutative symmetric functions is the free ring Z⟨Z1, Z2,...⟩ generated by non-commuting variables Z1, Z2, ...
The coproduct takes Zn to Σ Zi ⊗ Zn–i, where Z0 = 1 is the identity.
The counit takes Zi to 0 for i > 0 and takes Z0 = 1 to 1.

<h2 id="related-notions">Related notions</h2>
Michiel Hazewinkel showed<a class="footnote-ref" id="fnref:2" href="#fn:2">2</a> that a <a href="/facts/Hasse%E2%80%93Schmidt_derivation/A3xoT2AJ">Hasse–Schmidt derivation</a>

D
        :
        A
        →
        A
        [
        [
        t
        ]
        ]
      
    
    {\displaystyle D:A\to A[[t]]}

on a ring A is equivalent to an action of NSymm on A: the part 
 
 
 
 
 D
 
 i
 
 
 :
 A
 →
 A
 
 
 {\displaystyle D_{i}:A\to A}
 
 of D which picks the coefficient of 
 
 
 
 
 t
 
 i
 
 
 
 
 {\displaystyle t^{i}}
 
, is the action of the indeterminate Zi.

<h3>Relation to free Lie algebra</h3>
The element Σ Zntn is a <a href="/facts/Group-like_element/pvx52xS6">group-like element</a> of the Hopf algebra of formal <a href="/facts/Power_series/vYh6sTps">power series</a> over NSymm, so over the rationals its logarithm is primitive. The coefficients of its logarithm generate the free Lie algebra on a countable set of generators over the rationals. Over the rationals this identifies the Hopf algebra NSYmm with the universal enveloping algebra of the free Lie algebra.

<h2 id="references">References</h2>

<ol>
<li id="fn:1">Gelfand, Israel M.; Krob, Daniel; Lascoux, Alain; Leclerc, Bernard; Retakh, Vladimir S.; Thibon, Jean-Yves (1995), "Noncommutative symmetric functions", Advances in Mathematics, 112 (2): 218–348, arXiv:hep-th/9407124, doi:10.1006/aima.1995.1032, MR 1327096 <a href="/wiki/Israel_M._Gelfand" target="_blank">/wiki/Israel_M._Gelfand</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></li>
<li id="fn:2">Hazewinkel, Michiel (2012), "Hasse–Schmidt Derivations and the Hopf Algebra of Non-Commutative Symmetric Functions", Axioms, 1 (2): 149–154, arXiv:1110.6108, doi:10.3390/axioms1020149 <a href="/wiki/ArXiv_(identifier)" target="_blank">/wiki/ArXiv_(identifier)</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></li>
</ol>

Noncommutative symmetric function open-in-new

Noncommutative symmetric function