Namespaces
Variants
Actions

Difference between revisions of "Primitive element in a co-algebra"

From Encyclopedia of Mathematics
Jump to: navigation, search
(Importing text file)
 
m (tex encoded by computer)
Line 1: Line 1:
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110220/p1102201.png" /> be a [[Co-algebra|co-algebra]] over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110220/p1102202.png" />. An element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110220/p1102203.png" /> is called group like if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110220/p1102204.png" />. An element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110220/p1102205.png" /> is called primitive over the group-like element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110220/p1102206.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110220/p1102207.png" />, [[#References|[a2]]], p. 199. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110220/p1102208.png" /> be a bi-algebra (see [[Hopf algebra|Hopf algebra]]) and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110220/p1102209.png" /> be the set of primitive elements over the group-like element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110220/p11022010.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110220/p11022011.png" />, considered as a co-algebra. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110220/p11022012.png" /> becomes a [[Lie algebra|Lie algebra]] under the commutator bracket
+
<!--
 +
p1102201.png
 +
$#A+1 = 26 n = 0
 +
$#C+1 = 26 : ~/encyclopedia/old_files/data/P110/P.1100220 Primitive element in a co\AAhalgebra
 +
Automatically converted into TeX, above some diagnostics.
 +
Please remove this comment and the {{TEX|auto}} line below,
 +
if TeX found to be correct.
 +
-->
  
<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/p/p110/p110220/p11022013.png" /></td> </tr></table>
+
{{TEX|auto}}
 +
{{TEX|done}}
  
(using the multiplication of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110220/p11022014.png" />). This is the Lie algebra of primitive elements.
+
Let  $  ( C, \mu, \epsilon ) $
 +
be a [[Co-algebra|co-algebra]] over  $  k $.
 +
An element  $  x \in C $
 +
is called group like if  $  \mu ( x ) = g \otimes g $.
 +
An element  $  x \in C $
 +
is called primitive over the group-like element  $  g $
 +
if  $  \mu ( x ) = g \otimes x + x \otimes g $,
 +
[[#References|[a2]]], p. 199. Let  $  ( B,m,e, \mu, \epsilon ) $
 +
be a bi-algebra (see [[Hopf algebra|Hopf algebra]]) and let  $  P ( B ) $
 +
be the set of primitive elements over the group-like element  $  1 \in B $
 +
of  $  B $,
 +
considered as a co-algebra. Then  $  P ( B ) $
 +
becomes a [[Lie algebra|Lie algebra]] under the commutator bracket
  
For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110220/p11022015.png" /> a [[Field|field]] of characteristic zero, the functors <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110220/p11022016.png" />, the [[Universal enveloping algebra|universal enveloping algebra]] of the Lie algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110220/p11022017.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110220/p11022018.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110220/p11022019.png" /> is a Hopf algebra (or bi-algebra) over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110220/p11022020.png" />, establish an equivalence between the category of Lie algebras and the category of co-commutative irreducible bi-algebras (such bi-algebras are automatically Hopf algebras).
+
$$
 +
[ x,y ] = xy = yx,
 +
$$
  
In particular, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110220/p11022021.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110220/p11022022.png" /> for such a bi-algebra (Hopf algebra) [[#References|[a2]]], [[#References|[a1]]]; for the graded version of this correspondence, see [[Hopf algebra|Hopf algebra]] and the references quoted there. See also [[Lie polynomial|Lie polynomial]] for the concrete case that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110220/p11022023.png" /> is a free Lie algebra (cf. [[Lie algebra, free|Lie algebra, free]]) on a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110220/p11022024.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110220/p11022025.png" /> is the free associative algebra over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110220/p11022026.png" />.
+
(using the multiplication of  $  B $).
 +
This is the Lie algebra of primitive elements.
 +
 
 +
For  $  k $
 +
a [[Field|field]] of characteristic zero, the functors  $  L \mapsto U ( L ) $,
 +
the [[Universal enveloping algebra|universal enveloping algebra]] of the Lie algebra  $  L $,
 +
and  $  H \mapsto P ( H ) $,
 +
where  $  H $
 +
is a Hopf algebra (or bi-algebra) over  $  k $,
 +
establish an equivalence between the category of Lie algebras and the category of co-commutative irreducible bi-algebras (such bi-algebras are automatically Hopf algebras).
 +
 
 +
In particular, $  P ( U ( L ) ) \simeq L $,  
 +
$  U ( P ( H ) ) \simeq H $
 +
for such a bi-algebra (Hopf algebra) [[#References|[a2]]], [[#References|[a1]]]; for the graded version of this correspondence, see [[Hopf algebra|Hopf algebra]] and the references quoted there. See also [[Lie polynomial|Lie polynomial]] for the concrete case that $  L $
 +
is a free Lie algebra (cf. [[Lie algebra, free|Lie algebra, free]]) on a set $  X $
 +
and $  U ( L ) = { \mathop{\rm Ass} } ( X ) $
 +
is the free associative algebra over $  X $.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  E. Abe,  "Hopf algebras" , Cambridge Univ. Press  (1977)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  M.E. Sweedler,  "Hopf algebras" , Benjamin  (1963)</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  E. Abe,  "Hopf algebras" , Cambridge Univ. Press  (1977)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  M.E. Sweedler,  "Hopf algebras" , Benjamin  (1963)</TD></TR></table>

Revision as of 08:07, 6 June 2020


Let $ ( C, \mu, \epsilon ) $ be a co-algebra over $ k $. An element $ x \in C $ is called group like if $ \mu ( x ) = g \otimes g $. An element $ x \in C $ is called primitive over the group-like element $ g $ if $ \mu ( x ) = g \otimes x + x \otimes g $, [a2], p. 199. Let $ ( B,m,e, \mu, \epsilon ) $ be a bi-algebra (see Hopf algebra) and let $ P ( B ) $ be the set of primitive elements over the group-like element $ 1 \in B $ of $ B $, considered as a co-algebra. Then $ P ( B ) $ becomes a Lie algebra under the commutator bracket

$$ [ x,y ] = xy = yx, $$

(using the multiplication of $ B $). This is the Lie algebra of primitive elements.

For $ k $ a field of characteristic zero, the functors $ L \mapsto U ( L ) $, the universal enveloping algebra of the Lie algebra $ L $, and $ H \mapsto P ( H ) $, where $ H $ is a Hopf algebra (or bi-algebra) over $ k $, establish an equivalence between the category of Lie algebras and the category of co-commutative irreducible bi-algebras (such bi-algebras are automatically Hopf algebras).

In particular, $ P ( U ( L ) ) \simeq L $, $ U ( P ( H ) ) \simeq H $ for such a bi-algebra (Hopf algebra) [a2], [a1]; for the graded version of this correspondence, see Hopf algebra and the references quoted there. See also Lie polynomial for the concrete case that $ L $ is a free Lie algebra (cf. Lie algebra, free) on a set $ X $ and $ U ( L ) = { \mathop{\rm Ass} } ( X ) $ is the free associative algebra over $ X $.

References

[a1] E. Abe, "Hopf algebras" , Cambridge Univ. Press (1977)
[a2] M.E. Sweedler, "Hopf algebras" , Benjamin (1963)
How to Cite This Entry:
Primitive element in a co-algebra. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Primitive_element_in_a_co-algebra&oldid=16750
This article was adapted from an original article by M. Hazewinkel (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article