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Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z099/z099150/z0991501.png" /> be the (graded) free [[Lie algebra|Lie algebra]] on two generators over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z099/z099150/z0991502.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z099/z099150/z0991503.png" /> the graded free associative algebra on two generators over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z099/z099150/z0991504.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z099/z099150/z0991505.png" /> its completion with respect to the augmentation ideal (where both <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z099/z099150/z0991506.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z099/z099150/z0991507.png" /> have degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z099/z099150/z0991508.png" />). For each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z099/z099150/z0991509.png" /> without constant term, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z099/z099150/z09915010.png" /> denote the element
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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/z/z099/z099150/z09915011.png" /></td> </tr></table>
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of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z099/z099150/z09915012.png" />. Then there exist elements <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z099/z099150/z09915013.png" />, homogeneous of degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z099/z099150/z09915014.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z099/z099150/z09915015.png" />, homogeneous of degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z099/z099150/z09915016.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z099/z099150/z09915017.png" /> and of degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z099/z099150/z09915018.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z099/z099150/z09915019.png" />, in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z099/z099150/z09915020.png" /> which are Lie elements, i.e. they are in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z099/z099150/z09915021.png" />, and which are such that
+
Let  $  L( X, Y) $
 +
be the (graded) free [[Lie algebra|Lie algebra]] on two generators over  $  \mathbf Z $,  
 +
$  \mathop{\rm Ass} ( X, Y) $
 +
the graded free associative algebra on two generators over  $  \mathbf Z $
 +
and $  { \mathop{\rm Ass} } hat ( X, Y) $
 +
its completion with respect to the augmentation ideal (where both  $  X $
 +
and $  Y $
 +
have degree $  1 $).  
 +
For each  $  z \in \mathop{\rm Ass} ( X, Y) $
 +
without constant term, let  $  e ^ {z} $
 +
denote the element
  
<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/z/z099/z099150/z09915022.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a1)</td></tr></table>
+
$$
 +
e  ^ {z}  = 1 + z +
 +
\frac{z  ^ {2} }{2!}
 +
+
 +
\frac{z ^ {3} }{3!}
 +
+ \dots
 +
$$
  
<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/z/z099/z099150/z09915023.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a2)</td></tr></table>
+
of  $  { \mathop{\rm Ass} } hat ( X, Y) $.  
 +
Then there exist elements  $  c _ {n} ( X, Y) $,
 +
homogeneous of degree  $  n $,
 +
and  $  R _ {m} ( X, Y) $,
 +
homogeneous of degree  $  m $
 +
in  $  X $
 +
and of degree  $  n $
 +
in  $  Y $,
 +
in  $  \mathop{\rm Ass} ( X, Y) $
 +
which are Lie elements, i.e. they are in  $  L( X, Y) \subset  \mathop{\rm Ass} ( X, Y) $,
 +
and which are such that
  
Here the factors on the right-hand side are to be taken in the natural order for (a1), while in the case of (a2) the product is first taken over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z099/z099150/z09915024.png" /> and then over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z099/z099150/z09915025.png" />. The <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z099/z099150/z09915026.png" /> are recursively defined by:
+
$$ \tag{a1 }
 +
e  ^ {X} e  ^ {Y}  =  \prod _ {n \geq  1 } e ^
 +
{c _ {n} ( X, Y) / n! } ,
 +
$$
  
<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/z/z099/z099150/z09915027.png" /></td> </tr></table>
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$$ \tag{a2 }
 +
e  ^ {-} X e  ^ {-} Y e  ^ {X} e  ^ {Y}  = \prod _ { n= } 1 ^  \infty  \prod _ { m= } 1 ^  \infty  e ^ {R _ {m,n }  ( X, Y)/m!n! } .
 +
$$
  
<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/z/z099/z099150/z09915028.png" /></td> </tr></table>
+
Here the factors on the right-hand side are to be taken in the natural order for (a1), while in the case of (a2) the product is first taken over  $  m $
 +
and then over  $  n $.
 +
The  $  c _ {n} ( X, Y) $
 +
are recursively defined by:
  
These formulas find application in (combinatorial) group theory, algebraic topology and quantum theory, cf., e.g., [[#References|[a2]]]–[[#References|[a4]]]. For convergence results (for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z099/z099150/z09915029.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z099/z099150/z09915030.png" /> elements of a Banach algebra) concerning formula (a1) and for more general formulas cf., e.g., [[#References|[a2]]].
+
$$
 +
c _ {n} ( X, Y) =
 +
$$
 +
 
 +
$$
 +
= \
 +
\left .
 +
\frac{\partial  ^ {n} }{\partial  t  ^ {n} }
 +
\left ( e ^ {- t  ^ {n-} 1 c _ {n-} 1 / ( n- 1)! } \dots e ^ {- t  ^ {2} c _ {2} / 2! }
 +
e ^ {- t c _ {1} } e  ^ {tX} e  ^ {tY} \right ) \right | _ {t= 0 }  .
 +
$$
 +
 
 +
These formulas find application in (combinatorial) group theory, algebraic topology and quantum theory, cf., e.g., [[#References|[a2]]]–[[#References|[a4]]]. For convergence results (for $  X $
 +
and $  Y $
 +
elements of a Banach algebra) concerning formula (a1) and for more general formulas cf., e.g., [[#References|[a2]]].
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  H. Zassenhaus,  "Über Lie'schen Ringe mit Primzahlcharakteristik"  ''Abh. Math. Sem. Univ. Hamburg'' , '''13'''  (1940)  pp. 1–100</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  M. Suzuki,  "On the convergence of exponential operators - the Zassenhaus formula, BCH formula and systematic approximants"  ''Comm. Math. Phys.'' , '''57'''  (1977)  pp. 193–200</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  W. Magnus,  A. Karrass,  B. Solitar,  "Combinatorial group theory: presentations in terms of generators and relations" , Wiley (Interscience)  (1966)  pp. 412</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  H.J. Baues,  "Commutator calculus and groups of homotopy classes" , Cambridge Univ. Press  (1981)</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  H. Zassenhaus,  "Über Lie'schen Ringe mit Primzahlcharakteristik"  ''Abh. Math. Sem. Univ. Hamburg'' , '''13'''  (1940)  pp. 1–100</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  M. Suzuki,  "On the convergence of exponential operators - the Zassenhaus formula, BCH formula and systematic approximants"  ''Comm. Math. Phys.'' , '''57'''  (1977)  pp. 193–200</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  W. Magnus,  A. Karrass,  B. Solitar,  "Combinatorial group theory: presentations in terms of generators and relations" , Wiley (Interscience)  (1966)  pp. 412</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  H.J. Baues,  "Commutator calculus and groups of homotopy classes" , Cambridge Univ. Press  (1981)</TD></TR></table>

Revision as of 08:29, 6 June 2020


Let $ L( X, Y) $ be the (graded) free Lie algebra on two generators over $ \mathbf Z $, $ \mathop{\rm Ass} ( X, Y) $ the graded free associative algebra on two generators over $ \mathbf Z $ and $ { \mathop{\rm Ass} } hat ( X, Y) $ its completion with respect to the augmentation ideal (where both $ X $ and $ Y $ have degree $ 1 $). For each $ z \in \mathop{\rm Ass} ( X, Y) $ without constant term, let $ e ^ {z} $ denote the element

$$ e ^ {z} = 1 + z + \frac{z ^ {2} }{2!} + \frac{z ^ {3} }{3!} + \dots $$

of $ { \mathop{\rm Ass} } hat ( X, Y) $. Then there exist elements $ c _ {n} ( X, Y) $, homogeneous of degree $ n $, and $ R _ {m} ( X, Y) $, homogeneous of degree $ m $ in $ X $ and of degree $ n $ in $ Y $, in $ \mathop{\rm Ass} ( X, Y) $ which are Lie elements, i.e. they are in $ L( X, Y) \subset \mathop{\rm Ass} ( X, Y) $, and which are such that

$$ \tag{a1 } e ^ {X} e ^ {Y} = \prod _ {n \geq 1 } e ^ {c _ {n} ( X, Y) / n! } , $$

$$ \tag{a2 } e ^ {-} X e ^ {-} Y e ^ {X} e ^ {Y} = \prod _ { n= } 1 ^ \infty \prod _ { m= } 1 ^ \infty e ^ {R _ {m,n } ( X, Y)/m!n! } . $$

Here the factors on the right-hand side are to be taken in the natural order for (a1), while in the case of (a2) the product is first taken over $ m $ and then over $ n $. The $ c _ {n} ( X, Y) $ are recursively defined by:

$$ c _ {n} ( X, Y) = $$

$$ = \ \left . \frac{\partial ^ {n} }{\partial t ^ {n} } \left ( e ^ {- t ^ {n-} 1 c _ {n-} 1 / ( n- 1)! } \dots e ^ {- t ^ {2} c _ {2} / 2! } e ^ {- t c _ {1} } e ^ {tX} e ^ {tY} \right ) \right | _ {t= 0 } . $$

These formulas find application in (combinatorial) group theory, algebraic topology and quantum theory, cf., e.g., [a2][a4]. For convergence results (for $ X $ and $ Y $ elements of a Banach algebra) concerning formula (a1) and for more general formulas cf., e.g., [a2].

References

[a1] H. Zassenhaus, "Über Lie'schen Ringe mit Primzahlcharakteristik" Abh. Math. Sem. Univ. Hamburg , 13 (1940) pp. 1–100
[a2] M. Suzuki, "On the convergence of exponential operators - the Zassenhaus formula, BCH formula and systematic approximants" Comm. Math. Phys. , 57 (1977) pp. 193–200
[a3] W. Magnus, A. Karrass, B. Solitar, "Combinatorial group theory: presentations in terms of generators and relations" , Wiley (Interscience) (1966) pp. 412
[a4] H.J. Baues, "Commutator calculus and groups of homotopy classes" , Cambridge Univ. Press (1981)
How to Cite This Entry:
Zassenhaus formula. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Zassenhaus_formula&oldid=13224