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Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110100/l1101001.png" /> be the [[Free associative algebra|free associative algebra]] on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110100/l1101002.png" /> over the integers. Give <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110100/l1101003.png" /> a [[Hopf algebra|Hopf algebra]] structure by means of the following co-multiplication, augmentation, and antipode:
+
Let $\mathbf{Z}\langle Z \rangle$ be the [[free associative algebra]] on $Z = \{Z_1,Z_2,\ldots\}$ over the integers. Give $\mathbf{Z}\langle Z \rangle$ a [[Hopf algebra]] structure by means of the following co-multiplication, augmentation, and antipode:
 
+
$$
<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/l110100/l1101004.png" /></td> </tr></table>
+
\mu(Z_n) = \sum_{i+j=n\,;\,\,i,j \ge 0} Z_i \otimes Z_j
 
+
$$
 
where
 
where
 +
$$
 +
Z_0 = 1
 +
$$
 +
$$
 +
\epsilon(Z_n) = 0,\ \ n=1,2,\ldots
 +
$$
 +
$$
 +
\iota(Z_n) = \sum_{i_1+\cdots+i_k = n} (-1)^k Z_{i_1} \cdots Z_{i_k} \ ,
 +
$$
 +
where the sum is over all strings $i_1,\ldots,i_k$, $i_j \ge 1$, such that $i_1+\cdots+i_k = n$. This makes $\mathbf{Z}\langle Z \rangle$ a Hopf algebra, called the Leibniz–Hopf algebra. This Hopf algebra is important, e.g., in the theory of curves of non-commutative formal groups (see [[Formal group]]) [[#References|[a1]]], [[#References|[a2]]], [[#References|[a5]]]. Its commutative quotient $\mathbf{Z}[Z]$, with the same co-multiplication, is the underlying Hopf algebra of the (big) Witt vector functor $R \mapsto W(R)$ (see [[Witt vector]]) and it plays an important role in the classification theory of unipotent commutative algebraic groups and in the theory of commutative formal groups (amongst other things) [[#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/l110100/l1101005.png" /></td> </tr></table>
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The Leibniz–Hopf algebra $\mathbf{Z}\langle Z \rangle$ is free as a $\mathbf{Z}$-module and graded. Its graded dual is also a Hopf algebra, whose underlying algebra is the overlapping shuffle algebra $\mathrm{OSh}(\mathbf{N})$. As a $\mathbf Z$-module, $\mathrm{OSh}(\mathbf{N})$ is free with basis $\mathbf{N}^*$, the free monoid (see [[Free semi-group]]) of all words in the alphabet $\mathbf{N}$ with the duality pairing $\mathbf{Z}\langle Z \rangle \times \mathrm{OSh}(\mathbf{N}) \rightarrow \mathbf{Z}$ given by
 
+
$$
<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/l110100/l1101006.png" /></td> </tr></table>
+
\left({ w , Z_{i_1}\cdots Z_{i_r} }\right) = \begin{cases}1,& w = i_1\cdots i_r ,\\ 0 &\text{otherwise}.\end{cases}
 
+
$$
<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/l110100/l1101007.png" /></td> </tr></table>
 
 
 
where the sum is over all strings <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110100/l1101008.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110100/l1101009.png" />, such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110100/l11010010.png" />. This makes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110100/l11010011.png" /> a Hopf algebra, called the Leibniz–Hopf algebra. This Hopf algebra is important, e.g., in the theory of curves of non-commutative formal groups (see [[Formal group|Formal group]]) [[#References|[a1]]], [[#References|[a2]]], [[#References|[a5]]]. Its commutative quotient <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110100/l11010012.png" />, with the same co-multiplication, is the underlying Hopf algebra of the (big) Witt vector functor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110100/l11010013.png" /> (see [[Witt vector|Witt vector]]) and it plays an important role in the classification theory of unipotent commutative algebraic groups and in the theory of commutative formal groups (amongst other things) [[#References|[a3]]].
 
 
 
The Leibniz–Hopf algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110100/l11010014.png" /> is free as a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110100/l11010015.png" />-module and graded. Its graded dual is also a Hopf algebra, whose underlying algebra is the overlapping shuffle algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110100/l11010016.png" />. As a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110100/l11010017.png" />-module, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110100/l11010018.png" /> is free with basis <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110100/l11010019.png" />, the free monoid (see [[Free semi-group|Free semi-group]]) of all words in the alphabet <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110100/l11010020.png" /> with the duality pairing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110100/l11010021.png" /> given by
 
 
 
<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/l110100/l11010022.png" /></td> </tr></table>
 
  
The overlapping shuffle product of two such words <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110100/l11010023.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110100/l11010024.png" /> is equal to
+
The overlapping shuffle product of two such words $u=(a_1,\ldots,a_s)$, $v = (b_1,\ldots,b_t)$ is equal to
 
+
$$
<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/l110100/l11010025.png" /></td> </tr></table>
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a \times_{\mathrm{OSh}} b = \sum_{f,g} (c_1,\ldots,c_r)
 
+
$$
where the sum is over all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110100/l11010026.png" /> and pairs of order-preserving injective mappings <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110100/l11010027.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110100/l11010028.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110100/l11010029.png" />, and where
+
where the sum is over all $r \in \mathbf{N}$ and pairs of order-preserving injective mappings $f : \{1,\ldots,s\} \rightarrow \{1,\ldots,r\}$, $g : \{1,\ldots,t\} \rightarrow \{1,\ldots,r\}$ such that $\mathrm{im}(f) \cup \mathrm{im}(g) = \{1,\ldots,r\}$, and where
 
+
$$
<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/l110100/l11010030.png" /></td> </tr></table>
+
c_i = a_{f^{-1}(i)} + b_{g^{-1}(i)}\ ,\ \ \ i = 1,\ldots,r
 
+
$$
with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110100/l11010031.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110100/l11010032.png" />, and similarly for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110100/l11010033.png" />.
+
with $a_{f^{-1}(i)} = 0 $ if $f^{-1}(i) = \emptyset$ , and similarly for $b_{g^{-1}(i)}$.
  
 
For example,
 
For example,
 +
$$
 +
(a) \times_{\mathrm{OSh}} (b_1,b_2) = (a,b_1,b_2) + (b_1,a,b_2) + (b_1,b_2,a) + (a+b_1,b_2) + (b_1,a+b_2) \ .
 +
$$
 +
The terms of maximal length of the overlapping shuffle product form the shuffle product, see [[Shuffle 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/l110100/l11010034.png" /></td> </tr></table>
+
A word $w \in \mathbf{N}^*$, $w = (a_1,\ldots,a_s)$, is elementary if the greatest common divisor of $a_1,\ldots,a_s$ is $1$. With this terminology, the Ditters–Scholtens theorem [[#References|[a4]]], [[#References|[a5]]] says that, as an algebra over $\mathbf{Z}$, the overlapping shuffle algebra $\mathrm{Osh}(\mathbf{Z})$ is the free commutative polynomial algebra with as generators the elementary concatenation powers of elementary Lyndon words (see [[Lyndon word]]). (E.g., the third concatenation power of $(a_1,a_2)$ is $(a_1,a_2,a_1,a_2,a_1,a_2)$.) In contrast with the case of the [[shuffle algebra]], this theorem already holds over $\mathbf{Z}$ (not just over $\mathbf{Q}$).
  
<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/l110100/l11010035.png" /></td> </tr></table>
+
====References====
 +
<table>
 +
<TR><TD valign="top">[a1]</TD> <TD valign="top"> E.J. Ditters, "Curves and formal (co)groups" ''Invent. Math.'' , '''17''' (1972) pp. 1–20 {{ZBL|0253.22010}}</TD></TR>
 +
<TR><TD valign="top">[a2]</TD> <TD valign="top"> E.J. Ditters, "Groupes formels" , ''Lecture Notes'' , Univ. Paris XI: Orsay (1974)</TD></TR>
 +
<TR><TD valign="top">[a3]</TD> <TD valign="top"> M. Hazewinkel, "Formal groups and applications" , Acad. Press (1978) {{MR|0506881}} {{MR|0463184}} {{ZBL|0454.14020}} </TD></TR>
 +
<TR><TD valign="top">[a4]</TD> <TD valign="top"> M. Hazewinkel, "The Leibniz Hopf algebra and Lyndon words" ''Preprint AM CWI'' , '''9612''' (1996)</TD></TR>
 +
<TR><TD valign="top">[a5]</TD> <TD valign="top"> A.C.J. Scholtens, "$S$-Typical curves in non-commutative Hopf algebras" , Free Univ. Amsterdam (1996) (Thesis)</TD></TR>
 +
<TR><TD valign="top">[b1]</TD> <TD valign="top"> Michiel Hazewinkel, "The Algebra of Quasi-Symmetric Functions is Free over the Integers", ''Advances in Mathematics'' '''164''' (2001) 283–300 {{DOI|10.1006/aima.2001.2017}}</TD></TR>
 +
</table>
  
The terms of maximal length of the overlapping shuffle product form the shuffle product, see [[Shuffle algebra|Shuffle algebra]].
+
[[Category:TeX done]]
 
 
A word <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110100/l11010036.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110100/l11010037.png" />, is elementary if the greatest common divisor of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110100/l11010038.png" /> is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110100/l11010039.png" />. With this terminology, the Ditters–Scholtens theorem [[#References|[a4]]], [[#References|[a5]]] says that, as an algebra over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110100/l11010040.png" />, the overlapping shuffle algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110100/l11010041.png" /> is the free commutative polynomial algebra with as generators the elementary concatenation powers of elementary Lyndon words (see [[Lyndon word|Lyndon word]]). (E.g., the third concatenation power of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110100/l11010042.png" /> is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110100/l11010043.png" />.) In contrast with the case of the [[Shuffle algebra|shuffle algebra]], this theorem already holds over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110100/l11010044.png" /> (not just over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110100/l11010045.png" />).
 
 
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> E.J. Ditters, "Curves and formal (co)groups" ''Invent. Math.'' , '''17''' (1972) pp. 1–20</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> E.J. Ditters, "Groupes formels" , ''Lecture Notes'' , Univ. Paris XI: Orsay (1974)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> M. Hazewinkel, "Formal groups and applications" , Acad. Press (1978) {{MR|0506881}} {{MR|0463184}} {{ZBL|0454.14020}} </TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top"> M. Hazewinkel, "The Leibniz Hopf algebra and Lyndon words" ''Preprint AM CWI'' , '''9612''' (1996)</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top"> A.C.J. Scholtens, "<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110100/l11010046.png" />-Typical curves in non-commutative Hopf algebras" , Free Univ. Amsterdam (1996) (Thesis)</TD></TR></table>
 

Latest revision as of 13:46, 20 March 2023

Let $\mathbf{Z}\langle Z \rangle$ be the free associative algebra on $Z = \{Z_1,Z_2,\ldots\}$ over the integers. Give $\mathbf{Z}\langle Z \rangle$ a Hopf algebra structure by means of the following co-multiplication, augmentation, and antipode: $$ \mu(Z_n) = \sum_{i+j=n\,;\,\,i,j \ge 0} Z_i \otimes Z_j $$ where $$ Z_0 = 1 $$ $$ \epsilon(Z_n) = 0,\ \ n=1,2,\ldots $$ $$ \iota(Z_n) = \sum_{i_1+\cdots+i_k = n} (-1)^k Z_{i_1} \cdots Z_{i_k} \ , $$ where the sum is over all strings $i_1,\ldots,i_k$, $i_j \ge 1$, such that $i_1+\cdots+i_k = n$. This makes $\mathbf{Z}\langle Z \rangle$ a Hopf algebra, called the Leibniz–Hopf algebra. This Hopf algebra is important, e.g., in the theory of curves of non-commutative formal groups (see Formal group) [a1], [a2], [a5]. Its commutative quotient $\mathbf{Z}[Z]$, with the same co-multiplication, is the underlying Hopf algebra of the (big) Witt vector functor $R \mapsto W(R)$ (see Witt vector) and it plays an important role in the classification theory of unipotent commutative algebraic groups and in the theory of commutative formal groups (amongst other things) [a3].

The Leibniz–Hopf algebra $\mathbf{Z}\langle Z \rangle$ is free as a $\mathbf{Z}$-module and graded. Its graded dual is also a Hopf algebra, whose underlying algebra is the overlapping shuffle algebra $\mathrm{OSh}(\mathbf{N})$. As a $\mathbf Z$-module, $\mathrm{OSh}(\mathbf{N})$ is free with basis $\mathbf{N}^*$, the free monoid (see Free semi-group) of all words in the alphabet $\mathbf{N}$ with the duality pairing $\mathbf{Z}\langle Z \rangle \times \mathrm{OSh}(\mathbf{N}) \rightarrow \mathbf{Z}$ given by $$ \left({ w , Z_{i_1}\cdots Z_{i_r} }\right) = \begin{cases}1,& w = i_1\cdots i_r ,\\ 0 &\text{otherwise}.\end{cases} $$

The overlapping shuffle product of two such words $u=(a_1,\ldots,a_s)$, $v = (b_1,\ldots,b_t)$ is equal to $$ a \times_{\mathrm{OSh}} b = \sum_{f,g} (c_1,\ldots,c_r) $$ where the sum is over all $r \in \mathbf{N}$ and pairs of order-preserving injective mappings $f : \{1,\ldots,s\} \rightarrow \{1,\ldots,r\}$, $g : \{1,\ldots,t\} \rightarrow \{1,\ldots,r\}$ such that $\mathrm{im}(f) \cup \mathrm{im}(g) = \{1,\ldots,r\}$, and where $$ c_i = a_{f^{-1}(i)} + b_{g^{-1}(i)}\ ,\ \ \ i = 1,\ldots,r $$ with $a_{f^{-1}(i)} = 0 $ if $f^{-1}(i) = \emptyset$ , and similarly for $b_{g^{-1}(i)}$.

For example, $$ (a) \times_{\mathrm{OSh}} (b_1,b_2) = (a,b_1,b_2) + (b_1,a,b_2) + (b_1,b_2,a) + (a+b_1,b_2) + (b_1,a+b_2) \ . $$ The terms of maximal length of the overlapping shuffle product form the shuffle product, see Shuffle algebra.

A word $w \in \mathbf{N}^*$, $w = (a_1,\ldots,a_s)$, is elementary if the greatest common divisor of $a_1,\ldots,a_s$ is $1$. With this terminology, the Ditters–Scholtens theorem [a4], [a5] says that, as an algebra over $\mathbf{Z}$, the overlapping shuffle algebra $\mathrm{Osh}(\mathbf{Z})$ is the free commutative polynomial algebra with as generators the elementary concatenation powers of elementary Lyndon words (see Lyndon word). (E.g., the third concatenation power of $(a_1,a_2)$ is $(a_1,a_2,a_1,a_2,a_1,a_2)$.) In contrast with the case of the shuffle algebra, this theorem already holds over $\mathbf{Z}$ (not just over $\mathbf{Q}$).

References

[a1] E.J. Ditters, "Curves and formal (co)groups" Invent. Math. , 17 (1972) pp. 1–20 Zbl 0253.22010
[a2] E.J. Ditters, "Groupes formels" , Lecture Notes , Univ. Paris XI: Orsay (1974)
[a3] M. Hazewinkel, "Formal groups and applications" , Acad. Press (1978) MR0506881 MR0463184 Zbl 0454.14020
[a4] M. Hazewinkel, "The Leibniz Hopf algebra and Lyndon words" Preprint AM CWI , 9612 (1996)
[a5] A.C.J. Scholtens, "$S$-Typical curves in non-commutative Hopf algebras" , Free Univ. Amsterdam (1996) (Thesis)
[b1] Michiel Hazewinkel, "The Algebra of Quasi-Symmetric Functions is Free over the Integers", Advances in Mathematics 164 (2001) 283–300 DOI 10.1006/aima.2001.2017
How to Cite This Entry:
Leibniz-Hopf algebra. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Leibniz-Hopf_algebra&oldid=24030
This article was adapted from an original article by M. Hazewinkel (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article