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Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110130/l1101301.png" /> denote the [[Free associative algebra|free associative algebra]] over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110130/l1101302.png" /> in the indeterminates <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110130/l1101303.png" />. Give <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110130/l1101304.png" /> bi-algebra and [[Hopf algebra|Hopf algebra]] structures by means of the co-multiplication, antipode, and augmentation defined by:
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Let $\mathbf{Z}\langle X \rangle$ denote the [[free associative algebra]] over $\mathbf{Z}$ in the indeterminates $X = \{ X_i \ :\ i \in I \}$. Give $\mathbf{Z}\langle X \rangle$ bi-algebra and [[Hopf algebra|Hopf algebra]] structures by means of the co-multiplication, antipode, and augmentation defined by:
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$$
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\mu(X_i) = 1 \otimes X_i + X_i \otimes 1\ ,
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$$
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$$
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\epsilon(X_i) = 0 \ ,
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$$
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$$
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\iota(X_i) = -X_i \ .
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$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110130/l1101305.png" /></td> </tr></table>
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Then $\mathbf{Z}\langle X \rangle$ becomes the [[Leibniz–Hopf algebra]]. A Lie polynomial is an element $P$ of $\mathbf{Z}\langle X \rangle$ such that $\mu(P) = 1 \otimes P + P \otimes 1$, i.e., the Lie polynomials are the primitive elements of the Hopf algebra $\mathbf{Z}\langle X \rangle$ (see [[Primitive element in a co-algebra]]). These form a [[Lie algebra]] $L$ under the commutator difference product $[P,Q] = PQ - QP$. The Lie algebra $L$ is the free Lie algebra on $X$ over $\mathbf{Z}$ (Friedrich's theorem; cf. also [[Lie algebra, free]]) and $\mathbf{Z}\langle X \rangle$ is its [[universal enveloping algebra]].
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110130/l1101306.png" /></td> </tr></table>
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For bases of $L$ viewed as a submodule of $\mathbf{Z}\langle X \rangle$, see [[Hall set]]; [[Shirshov basis]]; [[Lyndon word]]. Still other bases, such as the Meier–Wunderli basis and the Spitzer–Foata basis, can be found in [[#References|[a3]]].
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110130/l1101307.png" /></td> </tr></table>
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====References====
 
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<table>
Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110130/l1101308.png" /> becomes the [[Leibniz–Hopf algebra|Leibniz–Hopf algebra]]. A Lie polynomial is an element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110130/l1101309.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110130/l11013010.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110130/l11013011.png" />, i.e., the Lie polynomials are the primitive elements of the Hopf algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110130/l11013012.png" /> (see [[Primitive element in a co-algebra|Primitive element in a co-algebra]]). These form a [[Lie algebra|Lie algebra]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110130/l11013013.png" /> under the commutator difference product <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110130/l11013014.png" />. The Lie algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110130/l11013015.png" /> is the free Lie algebra on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110130/l11013016.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110130/l11013017.png" /> (Friedrich's theorem; cf. also [[Lie algebra, free|Lie algebra, free]]) and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110130/l11013018.png" /> is its [[Universal enveloping algebra|universal enveloping algebra]].
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<TR><TD valign="top">[a1]</TD> <TD valign="top">  N. Bourbaki,  "Groupes de Lie" , '''II: Algèbres de Lie libres''' , Hermann  (1972)</TD></TR>
 
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<TR><TD valign="top">[a2]</TD> <TD valign="top">  C. Reutenauer,  "Free Lie algebras" , Oxford Univ. Press  (1993)</TD></TR>
For bases of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110130/l11013019.png" /> viewed as a submodule of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110130/l11013020.png" />, see [[Hall set|Hall set]]; [[Shirshov basis|Shirshov basis]]; [[Lyndon word|Lyndon word]]. Still other bases, such as the Meier–Wunderli basis and the Spitzer–Foata basis, can be found in [[#References|[a3]]].
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<TR><TD valign="top">[a3]</TD> <TD valign="top">  X. Viennot,  "Algèbres de Lie libres et monoïdes libres" , Springer  (1978)</TD></TR>
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<TR><TD valign="top">[a4]</TD> <TD valign="top">  J.-P. Serre,  "Lie algebras and Lie groups" , Benjamin  (1965)</TD></TR>
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</table>
  
====References====
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{{TEX|done}}
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  N. Bourbaki,  "Groupes de Lie" , '''II: Algèbres de Lie libres''' , Hermann  (1972)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  C. Reutenauer,  "Free Lie algebras" , Oxford Univ. Press  (1993)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  X. Viennot,  "Algèbres de Lie libres et monoïdes libres" , Springer  (1978)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  J.-P. Serre,  "Lie algebras and Lie groups" , Benjamin  (1965)</TD></TR></table>
 

Latest revision as of 20:59, 9 December 2015

Let $\mathbf{Z}\langle X \rangle$ denote the free associative algebra over $\mathbf{Z}$ in the indeterminates $X = \{ X_i \ :\ i \in I \}$. Give $\mathbf{Z}\langle X \rangle$ bi-algebra and Hopf algebra structures by means of the co-multiplication, antipode, and augmentation defined by: $$ \mu(X_i) = 1 \otimes X_i + X_i \otimes 1\ , $$ $$ \epsilon(X_i) = 0 \ , $$ $$ \iota(X_i) = -X_i \ . $$

Then $\mathbf{Z}\langle X \rangle$ becomes the Leibniz–Hopf algebra. A Lie polynomial is an element $P$ of $\mathbf{Z}\langle X \rangle$ such that $\mu(P) = 1 \otimes P + P \otimes 1$, i.e., the Lie polynomials are the primitive elements of the Hopf algebra $\mathbf{Z}\langle X \rangle$ (see Primitive element in a co-algebra). These form a Lie algebra $L$ under the commutator difference product $[P,Q] = PQ - QP$. The Lie algebra $L$ is the free Lie algebra on $X$ over $\mathbf{Z}$ (Friedrich's theorem; cf. also Lie algebra, free) and $\mathbf{Z}\langle X \rangle$ is its universal enveloping algebra.

For bases of $L$ viewed as a submodule of $\mathbf{Z}\langle X \rangle$, see Hall set; Shirshov basis; Lyndon word. Still other bases, such as the Meier–Wunderli basis and the Spitzer–Foata basis, can be found in [a3].

References

[a1] N. Bourbaki, "Groupes de Lie" , II: Algèbres de Lie libres , Hermann (1972)
[a2] C. Reutenauer, "Free Lie algebras" , Oxford Univ. Press (1993)
[a3] X. Viennot, "Algèbres de Lie libres et monoïdes libres" , Springer (1978)
[a4] J.-P. Serre, "Lie algebras and Lie groups" , Benjamin (1965)
How to Cite This Entry:
Lie polynomial. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Lie_polynomial&oldid=36882
This article was adapted from an original article by M. Hazewinkel (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article