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''bi-algebra, hyperalgebra''
 
''bi-algebra, hyperalgebra''
  
A graded module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047970/h0479701.png" /> over an associative-commutative ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047970/h0479702.png" /> with identity, equipped simultaneously with the structure of an associative graded algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047970/h0479703.png" /> with identity (unit element) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047970/h0479704.png" /> and the structure of an associative graded [[Co-algebra|co-algebra]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047970/h0479705.png" /> with co-identity (co-unit) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047970/h0479706.png" />, satisfying the following conditions:
+
A graded module $  A $
 +
over an associative-commutative ring $  K $
 +
with identity, equipped simultaneously with the structure of an associative graded algebra $  \mu : \  A \otimes A \rightarrow A $
 +
with identity (unit element) $  \iota : \  K \rightarrow A $
 +
and the structure of an associative graded [[Co-algebra|co-algebra]] $  \delta : \  A \rightarrow A \otimes A $
 +
with co-identity (co-unit) $  \epsilon : \  A \rightarrow K $ ,  
 +
satisfying the following conditions:
  
1) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047970/h0479707.png" /> is a homomorphism of graded co-algebras;
+
1) $  \iota $
 +
is a homomorphism of graded co-algebras;
  
2) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047970/h0479708.png" /> is a homomorphism of graded algebras;
+
2) $  \epsilon $
 +
is a homomorphism of graded algebras;
  
3) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047970/h0479709.png" /> is a homomorphism of graded algebras.
+
3) $  \delta $
 +
is a homomorphism of graded algebras.
  
 
Condition 3) is equivalent to:
 
Condition 3) is equivalent to:
  
3') <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047970/h04797010.png" /> is a homomorphism of graded co-algebras.
+
3') $  \mu $
 +
is a homomorphism of graded co-algebras.
  
 
Sometimes the requirement that the co-multiplication is associative is discarded; such algebras are called quasi-Hopf algebras.
 
Sometimes the requirement that the co-multiplication is associative is discarded; such algebras are called quasi-Hopf algebras.
  
For any two Hopf algebras <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047970/h04797011.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047970/h04797012.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047970/h04797013.png" /> their tensor product <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047970/h04797014.png" /> is endowed with the natural structure of a Hopf algebra. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047970/h04797015.png" /> be a Hopf algebra, where all the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047970/h04797016.png" /> are finitely-generated projective <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047970/h04797017.png" />-modules. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047970/h04797018.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047970/h04797019.png" /> is the module dual to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047970/h04797020.png" />, endowed with the homomorphisms of graded modules <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047970/h04797021.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047970/h04797022.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047970/h04797023.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047970/h04797024.png" />, is a Hopf algebra; it is said to be dual to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047970/h04797025.png" />. An element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047970/h04797026.png" /> of a Hopf algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047970/h04797027.png" /> is called primitive if
+
For any two Hopf algebras $  A $
 
+
and $  B $
<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/h/h047/h047970/h04797028.png" /></td> </tr></table>
+
over $  K $
 
+
their tensor product $  A \otimes B $
The primitive elements form a graded subalgebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047970/h04797029.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047970/h04797030.png" /> under the operation
+
is endowed with the natural structure of a Hopf algebra. Let $  A = \sum _ {n \in \mathbf Z} A _{n} $
 
+
be a Hopf algebra, where all the $  A _{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/h/h047/h047970/h04797031.png" /></td> </tr></table>
+
are finitely-generated projective $  K $ -
 
+
modules. Then $  A ^{*} = \sum _ {n \in \mathbf Z} A _{n} ^{*} $ ,  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047970/h04797032.png" /> is connected (that is, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047970/h04797033.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047970/h04797034.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047970/h04797035.png" />) and if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047970/h04797036.png" /> is a field of characteristic 0, then the subspace <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047970/h04797037.png" /> generates the algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047970/h04797038.png" /> (with respect to multiplication) if and only if the co-multiplication is graded commutative [[#References|[2]]].
+
where $  A _{n} ^{*} $
 +
is the module dual to $  A _{n} $ ,  
 +
endowed with the homomorphisms of graded modules $  \delta ^{*} : \  A ^{*} \otimes A ^{*} \rightarrow A ^{*} $ ,  
 +
$  \epsilon ^{*} : \  K \rightarrow A ^{*} $ ,  
 +
$  \mu ^{*} : \  A ^{*} \rightarrow A ^{*} \otimes A ^{*} $ ,  
 +
$  \iota ^{*} : \  A ^{*} \rightarrow K $ ,  
 +
is a Hopf algebra; it is said to be dual to $  A $ .  
 +
An element $  x $
 +
of a Hopf algebra $  A $
 +
is called primitive if$$
 +
\delta (x)  = 
 +
x \otimes 1 + 1 \otimes x.
 +
$$
 +
The primitive elements form a graded subalgebra $  P _{A} $
 +
in $  A $
 +
under the operation$$
 +
[x,\  y]  =   xy - (-1) ^{pq} yx, 
 +
x \in A _{p} ,  y \in A _{q} .
 +
$$
 +
If $  A $
 +
is connected (that is, $  A _{n} = 0 $
 +
for $  n < 0 $ ,  
 +
$  A _{0} = K \  $ )  
 +
and if $  K $
 +
is a field of characteristic 0, then the subspace $  P _{A} $
 +
generates the algebra $  A $ (
 +
with respect to multiplication) if and only if the co-multiplication is graded commutative [[#References|[2]]].
  
 
===Examples.===
 
===Examples.===
  
  
1) For any graded Lie algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047970/h04797039.png" /> (that is, a graded algebra that is a Lie [[Superalgebra|superalgebra]] under the natural <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047970/h04797040.png" />-grading) the universal enveloping algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047970/h04797041.png" /> becomes a Hopf algebra if one puts
+
1) For any graded Lie algebra $  \mathfrak g $ (
 +
that is, a graded algebra that is a Lie [[Superalgebra|superalgebra]] under the natural $  \mathbf Z _{2} $ -
 +
grading) the universal enveloping algebra $  U ( \mathfrak g ) $
 +
becomes a Hopf algebra if one puts$$
 +
\epsilon (x)  =   0
 +
\delta (x)  =   x \otimes 1 + 1 \otimes x, 
 +
x \in \mathfrak g .
 +
$$
 +
Here $  P _ {U ( \mathfrak g )} = \mathfrak g $ .  
 +
If $  K $
 +
is a field of characteristic 0, then any connected Hopf algebra $  A $
 +
generated by primitive elements is naturally isomorphic to $  U (P _{A} ) $ (
 +
see [[#References|[2]]]).
  
<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/h/h047/h047970/h04797042.png" /></td> </tr></table>
+
2) Similarly, the structure of a Hopf algebra (with a trivial grading) is defined in the group algebra $  K [G] $
 +
of an arbitrary group $  G $ .
  
Here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047970/h04797043.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047970/h04797044.png" /> is a field of characteristic 0, then any connected Hopf algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047970/h04797045.png" /> generated by primitive elements is naturally isomorphic to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047970/h04797046.png" /> (see [[#References|[2]]]).
 
  
2) Similarly, the structure of a Hopf algebra (with a trivial grading) is defined in the group algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047970/h04797047.png" /> of an arbitrary group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047970/h04797048.png" />.
+
3) The algebra of regular functions on an affine algebraic group $  G $
 +
becomes a Hopf algebra (with trivial grading) if one defines the homomorphisms $  \delta $
 +
and $  \epsilon $
 +
by means of the multiplication $  G \times G \rightarrow G $
 +
and the imbedding $  \{ e \} \rightarrow G $ ,
 +
where $  e $
 +
is the unit element of $  G $ (
 +
see [[#References|[3]]]).
  
3) The algebra of regular functions on an affine algebraic group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047970/h04797049.png" /> becomes a Hopf algebra (with trivial grading) if one defines the homomorphisms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047970/h04797050.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047970/h04797051.png" /> by means of the multiplication <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047970/h04797052.png" /> and the imbedding <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047970/h04797053.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047970/h04797054.png" /> is the unit element of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047970/h04797055.png" /> (see [[#References|[3]]]).
+
4) Suppose that $  G $
 +
is a path-connected [[H-space|$  H $ -
 +
space]] with multiplication $  m $
 +
and unit element $  e $
 +
and suppose that $  \Delta : \  G \rightarrow G \times G $ ,
 +
$  \iota : \  \{ e \} \rightarrow G $ ,
 +
$  p: \  G \rightarrow \{ e \} $
 +
are defined by the formulas $  \Delta (a) = (a,\  a) $ ,
 +
$  \iota (e) = e $ ,
 +
$  p (a) = e $ ,
 +
$  a \in G $ .  
 +
If all cohomology modules $  H ^{n} (G,\  K) $
 +
are projective and finitely generated, then the mappings $  \mu = \Delta ^{*} $ ,
 +
$  \iota = p ^{*} $ ,
 +
$  \delta = m ^{*} $ ,
 +
$  \epsilon = \iota ^{*} $
 +
induced in the cohomology, turn $  H ^{*} (G,\  K) $
 +
into a graded commutative quasi-Hopf algebra. If the multiplication $  m $
 +
is homotopy-associative, then $  H ^{*} (G ,\  K) $
 +
is a Hopf algebra, and the Hopf algebra dual to it is the homology algebra $  H _{*} (G,\  K) $ ,
 +
equipped with the mappings $  m _{*} $ ,
 +
$  \iota _{*} $ ,
 +
$  \Delta _{*} $ ,
 +
$  p _{*} $ (
 +
the Pontryagin algebra). If $  K $
 +
is a field of characteristic 0, then the Pontryagin algebra is generated by primitive elements and is isomorphic to $  U ( \pi (G,\  K)) $ ,
 +
where $  \pi (G,\  K) = \sum _ {i = 0} ^ \infty  \pi _{i} (G) \otimes K $
 +
is regarded as a graded Lie algebra under the Samelson product (see [[#References|[2]]]).
  
4) Suppose that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047970/h04797056.png" /> is a path-connected [[H-space|<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047970/h04797057.png" />-space]] with multiplication <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047970/h04797058.png" /> and unit element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047970/h04797059.png" /> and suppose that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047970/h04797060.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047970/h04797061.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047970/h04797062.png" /> are defined by the formulas <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047970/h04797063.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047970/h04797064.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047970/h04797065.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047970/h04797066.png" />. If all cohomology modules <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047970/h04797067.png" /> are projective and finitely generated, then the mappings <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047970/h04797068.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047970/h04797069.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047970/h04797070.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047970/h04797071.png" /> induced in the cohomology, turn <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047970/h04797072.png" /> into a graded commutative quasi-Hopf algebra. If the multiplication <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047970/h04797073.png" /> is homotopy-associative, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047970/h04797074.png" /> is a Hopf algebra, and the Hopf algebra dual to it is the homology algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047970/h04797075.png" />, equipped with the mappings <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047970/h04797076.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047970/h04797077.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047970/h04797078.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047970/h04797079.png" /> (the Pontryagin algebra). If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047970/h04797080.png" /> is a field of characteristic 0, then the Pontryagin algebra is generated by primitive elements and is isomorphic to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047970/h04797081.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047970/h04797082.png" /> is regarded as a graded Lie algebra under the Samelson product (see [[#References|[2]]]).
+
The algebra $  H ^{*} (G,\  K) $
 +
in Example 4) was first considered by H. Hopf in [[#References|[1]]], who showed that it is an exterior algebra with generators of odd degrees if $  K $
 +
is a field of characteristic 0 and $  H ^{*} (G,\  K) $
 +
is finite-dimensional. The structure of an arbitrary connected, graded, commutative quasi-Hopf algebra $  A $
 +
subject to the condition $  \mathop{\rm dim}\nolimits \  A _{n} < \infty $ ,  
 +
$  n \in \mathbf Z $ ,  
 +
over a perfect field $  K $
 +
of characteristic $  p $
 +
is described by the following theorem (see [[#References|[4]]]). The algebra $  A $
 +
splits into the tensor product of algebras with a single generator $  x $
 +
and the relation $  x ^{s} = 0 $ ,
 +
where for $  p = 2 $ ,  
 +
$  s $
 +
is a power of 2 or $  \infty $ ,  
 +
and for $  p \neq 2 $ ,
 +
$  s $
 +
is a power of $  p $
 +
or $  \infty $ (
 +
$  \infty $
 +
for $  p = 0 $ )
 +
if $  x $
 +
has even degree, and $  s = 2 $
 +
if the degree of $  x $
 +
is odd. In particular, for $  p = 0 $ ,  
 +
$  A $
 +
is the tensor product of an exterior algebra with generators of odd degree and an algebra of polynomials with generators of even degrees. On the other hand, every connected Hopf algebra $  A $
 +
over a field $  K $
 +
in which $  x ^{2} = 0 $
 +
for any element $  x $
 +
of odd degree and in which all elements of odd degree and all elements of even degree are decomposable, is the exterior algebra $  A = \land P _{A} $ (
 +
see [[#References|[2]]]). In particular, such are the cohomology algebra and the Pontryagin algebra of a connected compact Lie group over $  \mathbf R $ .
  
The algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047970/h04797083.png" /> in Example 4) was first considered by H. Hopf in [[#References|[1]]], who showed that it is an exterior algebra with generators of odd degrees if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047970/h04797084.png" /> is a field of characteristic 0 and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047970/h04797085.png" /> is finite-dimensional. The structure of an arbitrary connected, graded, commutative quasi-Hopf algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047970/h04797086.png" /> subject to the condition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047970/h04797087.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047970/h04797088.png" />, over a perfect field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047970/h04797089.png" /> of characteristic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047970/h04797090.png" /> is described by the following theorem (see [[#References|[4]]]). The algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047970/h04797091.png" /> splits into the tensor product of algebras with a single generator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047970/h04797092.png" /> and the relation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047970/h04797093.png" />, where for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047970/h04797094.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047970/h04797095.png" /> is a power of 2 or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047970/h04797096.png" />, and for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047970/h04797097.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047970/h04797098.png" /> is a power of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047970/h04797099.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047970/h047970100.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047970/h047970101.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047970/h047970102.png" />) if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047970/h047970103.png" /> has even degree, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047970/h047970104.png" /> if the degree of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047970/h047970105.png" /> is odd. In particular, for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047970/h047970106.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047970/h047970107.png" /> is the tensor product of an exterior algebra with generators of odd degree and an algebra of polynomials with generators of even degrees. On the other hand, every connected Hopf algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047970/h047970108.png" /> over a field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047970/h047970109.png" /> in which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047970/h047970110.png" /> for any element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047970/h047970111.png" /> of odd degree and in which all elements of odd degree and all elements of even degree are decomposable, is the exterior algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047970/h047970112.png" /> (see [[#References|[2]]]). In particular, such are the cohomology algebra and the Pontryagin algebra of a connected compact Lie group over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047970/h047970113.png" />.
 
  
 
====References====
 
====References====
Line 50: Line 162:
 
Terminology concerning Hopf algebras and bi-algebras is not yet quite standardized. However, the following nomenclature (and notation) seems to be on the way of being universally accepted.
 
Terminology concerning Hopf algebras and bi-algebras is not yet quite standardized. However, the following nomenclature (and notation) seems to be on the way of being universally accepted.
  
A bi-algebra is a module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047970/h047970114.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047970/h047970115.png" /> equipped with module mappings <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047970/h047970116.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047970/h047970117.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047970/h047970118.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047970/h047970119.png" /> such that
+
A bi-algebra is a module $  A $
 +
over $  K $
 +
equipped with module mappings $  m: \  A \otimes A \rightarrow A $ ,  
 +
$  e : \  K \rightarrow A $ ,  
 +
$  \mu : \  A \rightarrow A \otimes A $ ,  
 +
$  \epsilon : \  A \rightarrow K $
 +
such that
  
i) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047970/h047970120.png" /> is an associative algebra with unit;
+
i) $  ( A ,\  m ,\  e ) $
 +
is an associative algebra with unit;
  
ii) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047970/h047970121.png" /> is a co-associative co-algebra with co-unit;
+
ii) $  ( A ,\  \mu ,\  \epsilon ) $
 +
is a co-associative co-algebra with co-unit;
  
iii) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047970/h047970122.png" /> is a homomorphism of co-algebras;
+
iii) $  e $
 +
is a homomorphism of co-algebras;
  
iv) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047970/h047970123.png" /> is a homomorphism of algebras;
+
iv) $  \epsilon $
 +
is a homomorphism of algebras;
  
v) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047970/h047970124.png" /> is a homomorphism of co-algebras.
+
v) $  m $
 +
is a homomorphism of co-algebras.
  
 
This last condition is equivalent to:
 
This last condition is equivalent to:
  
v') <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047970/h047970125.png" /> is a homomorphism of algebras.
+
v') $  \mu $
 +
is a homomorphism of algebras.
  
 
A grading is not assumed to be part of the definition. If there is a grading and every morphism under consideration is graded, then one speaks of a graded bi-algebra.
 
A grading is not assumed to be part of the definition. If there is a grading and every morphism under consideration is graded, then one speaks of a graded bi-algebra.
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047970/h047970126.png" /> be a bi-algebra over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047970/h047970127.png" />. An antipode for the bi-algebra is a module homomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047970/h047970128.png" /> such that
+
Let $  ( A ,\  m ,\  e ,\  \mu ,\  \epsilon ) $
 +
be a bi-algebra over $  K $ .  
 +
An antipode for the bi-algebra is a module homomorphism $  \iota : \  A \rightarrow A $
 +
such that
  
vi) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047970/h047970129.png" />.
+
vi) $  m \circ ( \iota \otimes 1 ) \circ \mu = m \circ ( 1 \otimes \iota ) \circ \mu = e \circ \epsilon $ .
  
A bi-algebra with antipode <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047970/h047970130.png" /> is called a Hopf algebra. A graded Hopf algebra is a graded bi-algebra with antipode <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047970/h047970131.png" /> which is a homomorphism of graded modules.
 
  
Given a co-algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047970/h047970132.png" /> and an algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047970/h047970133.png" />, the module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047970/h047970134.png" /> admits a convolution product, defined as follows
+
A bi-algebra with antipode $  \iota $
 +
is called a Hopf algebra. A graded Hopf algebra is a graded bi-algebra with antipode $  \iota $
 +
which is a homomorphism of graded modules.
  
<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/h/h047/h047970/h047970135.png" /></td> </tr></table>
+
Given a co-algebra $  ( C ,\  \mu _{C} ,\  \epsilon _{C} ) $
 +
and an algebra $  ( A ,\  m _{A} ,\  e _{A} ) $ ,
 +
the module $  \mathop{\rm Mod}\nolimits _{K} ( C ,\  A ) $
 +
admits a convolution product, defined as follows$$
 +
f \star g  =   m _{A} \circ ( f \otimes g ) \circ \mu _{C} .
 +
$$
 +
In terms of this convolution product conditions vi) can be stated as
 +
 
 +
vi') $  \iota \star  \mathop{\rm id}\nolimits = \mathop{\rm id}\nolimits \star \iota = e \circ \epsilon $ ,
  
In terms of this convolution product conditions vi) can be stated as
 
  
vi') <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047970/h047970136.png" />,
+
where $  \mathop{\rm id}\nolimits : \  A \rightarrow A $
 +
is the identity morphism of the bi-algebra $  A $ .
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047970/h047970137.png" /> is the identity morphism of the bi-algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047970/h047970138.png" />.
 
  
An additional example of a Hopf algebra is the following. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047970/h047970139.png" /> be a [[Formal group|formal group]]. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047970/h047970140.png" />. Identifying <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047970/h047970141.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047970/h047970142.png" />, the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047970/h047970143.png" /> define a (continuous) algebra morphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047970/h047970144.png" /> turning <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047970/h047970145.png" /> into a bi-algebra. There is an antipode making <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047970/h047970146.png" /> a Hopf algebra. It is called the contravariant bi-algebra or contravariant Hopf algebra of the formal group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047970/h047970147.png" />. Note that here the completed tensor product is used.
+
An additional example of a Hopf algebra is the following. Let $  F _{1} ( X ; \  Y ) \dots F _{n} ( X ; \  Y ) \in K [ [ X _{1} \dots X _{n} ; \  Y _{1} \dots Y _{n} ] ] $
 +
be a [[Formal group|formal group]]. Let $  A = K [ [ X _{1} \dots X _{n} ] ] $ .  
 +
Identifying $  Y _{i} $
 +
with $  1 \otimes X _{i} \in A \widehat \otimes  A $ ,  
 +
the $  F _{1} \dots F _{n} $
 +
define a (continuous) algebra morphism $  \mu : \  A \rightarrow A \widehat \otimes  A $
 +
turning $  A $
 +
into a bi-algebra. There is an antipode making $  A $
 +
a Hopf algebra. It is called the contravariant bi-algebra or contravariant Hopf algebra of the formal group $  F $ .  
 +
Note that here the completed tensor product is used.
  
 
Hopf algebras, under the name quantum groups, and related objects have also become important in physics; in particular in connection with the quantum inverse-scattering method [[#References|[a3]]], [[#References|[a4]]].
 
Hopf algebras, under the name quantum groups, and related objects have also become important in physics; in particular in connection with the quantum inverse-scattering method [[#References|[a3]]], [[#References|[a4]]].
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> E. Abe, "Hopf algebras" , Cambridge Univ. Press (1977) {{MR|1857062}} {{MR|0594432}} {{MR|0321962}} {{ZBL|0476.16008}} {{ZBL|0236.14021}} </TD></TR><TR><TD valign="top">[a2]</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">[a3]</TD> <TD valign="top"> V.G. Drinfel'd, "Quantum groups" , ''Proc. Internat. Congress Mathematicians (Berkeley, 1986)'' , '''1''' , Amer. Math. Soc. (1987) pp. 798–820 {{MR|}} {{ZBL|0667.16003}} </TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top"> L.D. Faddeev, "Integrable models in (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047970/h047970148.png" />)-dimensional quantum field theory (Les Houches, 1982)" , Elsevier (1984) {{MR|782509}} {{ZBL|}} </TD></TR></table>
+
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> E. Abe, "Hopf algebras" , Cambridge Univ. Press (1977) {{MR|1857062}} {{MR|0594432}} {{MR|0321962}} {{ZBL|0476.16008}} {{ZBL|0236.14021}} </TD></TR><TR><TD valign="top">[a2]</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">[a3]</TD> <TD valign="top"> V.G. Drinfel'd, "Quantum groups" , ''Proc. Internat. Congress Mathematicians (Berkeley, 1986)'' , '''1''' , Amer. Math. Soc. (1987) pp. 798–820 {{MR|}} {{ZBL|0667.16003}} </TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top"> L.D. Faddeev, "Integrable models in (h047970148.png)-dimensional quantum field theory (Les Houches, 1982)" , Elsevier (1984) {{MR|782509}} {{ZBL|}} </TD></TR></table>

Latest revision as of 08:51, 16 December 2019

bi-algebra, hyperalgebra

A graded module $ A $ over an associative-commutative ring $ K $ with identity, equipped simultaneously with the structure of an associative graded algebra $ \mu : \ A \otimes A \rightarrow A $ with identity (unit element) $ \iota : \ K \rightarrow A $ and the structure of an associative graded co-algebra $ \delta : \ A \rightarrow A \otimes A $ with co-identity (co-unit) $ \epsilon : \ A \rightarrow K $ , satisfying the following conditions:

1) $ \iota $ is a homomorphism of graded co-algebras;

2) $ \epsilon $ is a homomorphism of graded algebras;

3) $ \delta $ is a homomorphism of graded algebras.

Condition 3) is equivalent to:

3') $ \mu $ is a homomorphism of graded co-algebras.

Sometimes the requirement that the co-multiplication is associative is discarded; such algebras are called quasi-Hopf algebras.

For any two Hopf algebras $ A $ and $ B $ over $ K $ their tensor product $ A \otimes B $ is endowed with the natural structure of a Hopf algebra. Let $ A = \sum _ {n \in \mathbf Z} A _{n} $ be a Hopf algebra, where all the $ A _{n} $ are finitely-generated projective $ K $ - modules. Then $ A ^{*} = \sum _ {n \in \mathbf Z} A _{n} ^{*} $ , where $ A _{n} ^{*} $ is the module dual to $ A _{n} $ , endowed with the homomorphisms of graded modules $ \delta ^{*} : \ A ^{*} \otimes A ^{*} \rightarrow A ^{*} $ , $ \epsilon ^{*} : \ K \rightarrow A ^{*} $ , $ \mu ^{*} : \ A ^{*} \rightarrow A ^{*} \otimes A ^{*} $ , $ \iota ^{*} : \ A ^{*} \rightarrow K $ , is a Hopf algebra; it is said to be dual to $ A $ . An element $ x $ of a Hopf algebra $ A $ is called primitive if$$ \delta (x) = x \otimes 1 + 1 \otimes x. $$ The primitive elements form a graded subalgebra $ P _{A} $ in $ A $ under the operation$$ [x,\ y] = xy - (-1) ^{pq} yx, x \in A _{p} , y \in A _{q} . $$ If $ A $ is connected (that is, $ A _{n} = 0 $ for $ n < 0 $ , $ A _{0} = K \ $ ) and if $ K $ is a field of characteristic 0, then the subspace $ P _{A} $ generates the algebra $ A $ ( with respect to multiplication) if and only if the co-multiplication is graded commutative [2].

Examples.

1) For any graded Lie algebra $ \mathfrak g $ ( that is, a graded algebra that is a Lie superalgebra under the natural $ \mathbf Z _{2} $ - grading) the universal enveloping algebra $ U ( \mathfrak g ) $ becomes a Hopf algebra if one puts$$ \epsilon (x) = 0, \delta (x) = x \otimes 1 + 1 \otimes x, x \in \mathfrak g . $$ Here $ P _ {U ( \mathfrak g )} = \mathfrak g $ . If $ K $ is a field of characteristic 0, then any connected Hopf algebra $ A $ generated by primitive elements is naturally isomorphic to $ U (P _{A} ) $ ( see [2]).

2) Similarly, the structure of a Hopf algebra (with a trivial grading) is defined in the group algebra $ K [G] $ of an arbitrary group $ G $ .


3) The algebra of regular functions on an affine algebraic group $ G $ becomes a Hopf algebra (with trivial grading) if one defines the homomorphisms $ \delta $ and $ \epsilon $ by means of the multiplication $ G \times G \rightarrow G $ and the imbedding $ \{ e \} \rightarrow G $ , where $ e $ is the unit element of $ G $ ( see [3]).

4) Suppose that $ G $ is a path-connected $ H $ - space with multiplication $ m $ and unit element $ e $ and suppose that $ \Delta : \ G \rightarrow G \times G $ , $ \iota : \ \{ e \} \rightarrow G $ , $ p: \ G \rightarrow \{ e \} $ are defined by the formulas $ \Delta (a) = (a,\ a) $ , $ \iota (e) = e $ , $ p (a) = e $ , $ a \in G $ . If all cohomology modules $ H ^{n} (G,\ K) $ are projective and finitely generated, then the mappings $ \mu = \Delta ^{*} $ , $ \iota = p ^{*} $ , $ \delta = m ^{*} $ , $ \epsilon = \iota ^{*} $ induced in the cohomology, turn $ H ^{*} (G,\ K) $ into a graded commutative quasi-Hopf algebra. If the multiplication $ m $ is homotopy-associative, then $ H ^{*} (G ,\ K) $ is a Hopf algebra, and the Hopf algebra dual to it is the homology algebra $ H _{*} (G,\ K) $ , equipped with the mappings $ m _{*} $ , $ \iota _{*} $ , $ \Delta _{*} $ , $ p _{*} $ ( the Pontryagin algebra). If $ K $ is a field of characteristic 0, then the Pontryagin algebra is generated by primitive elements and is isomorphic to $ U ( \pi (G,\ K)) $ , where $ \pi (G,\ K) = \sum _ {i = 0} ^ \infty \pi _{i} (G) \otimes K $ is regarded as a graded Lie algebra under the Samelson product (see [2]).

The algebra $ H ^{*} (G,\ K) $ in Example 4) was first considered by H. Hopf in [1], who showed that it is an exterior algebra with generators of odd degrees if $ K $ is a field of characteristic 0 and $ H ^{*} (G,\ K) $ is finite-dimensional. The structure of an arbitrary connected, graded, commutative quasi-Hopf algebra $ A $ subject to the condition $ \mathop{\rm dim}\nolimits \ A _{n} < \infty $ , $ n \in \mathbf Z $ , over a perfect field $ K $ of characteristic $ p $ is described by the following theorem (see [4]). The algebra $ A $ splits into the tensor product of algebras with a single generator $ x $ and the relation $ x ^{s} = 0 $ , where for $ p = 2 $ , $ s $ is a power of 2 or $ \infty $ , and for $ p \neq 2 $ , $ s $ is a power of $ p $ or $ \infty $ ( $ \infty $ for $ p = 0 $ ) if $ x $ has even degree, and $ s = 2 $ if the degree of $ x $ is odd. In particular, for $ p = 0 $ , $ A $ is the tensor product of an exterior algebra with generators of odd degree and an algebra of polynomials with generators of even degrees. On the other hand, every connected Hopf algebra $ A $ over a field $ K $ in which $ x ^{2} = 0 $ for any element $ x $ of odd degree and in which all elements of odd degree and all elements of even degree are decomposable, is the exterior algebra $ A = \land P _{A} $ ( see [2]). In particular, such are the cohomology algebra and the Pontryagin algebra of a connected compact Lie group over $ \mathbf R $ .


References

[1] H. Hopf, "Ueber die Topologie der Gruppenmannigfaltigkeiten und ihrer Verallgemeinerungen" Ann. of Math. (2) , 42 (1941) pp. 22–52
[2] J.W. Milnor, J.C. Moore, "On the structure of Hopf algebras" Ann. of Math. (2) , 81 : 2 (1965) pp. 211–264 MR0174052 Zbl 0163.28202
[3] A. Borel, "Linear algebraic groups" , Benjamin (1969) MR0251042 Zbl 0206.49801 Zbl 0186.33201
[4] A. Borel, "Sur la cohomologie des espaces fibrés principaux et des espaces homogènes de groupes de Lie compacts" Ann. of Math. , 57 (1953) pp. 115–207 MR0051508 Zbl 0052.40001
[5] S. MacLane, "Homology" , Springer (1963) Zbl 0818.18001 Zbl 0328.18009


Comments

Terminology concerning Hopf algebras and bi-algebras is not yet quite standardized. However, the following nomenclature (and notation) seems to be on the way of being universally accepted.

A bi-algebra is a module $ A $ over $ K $ equipped with module mappings $ m: \ A \otimes A \rightarrow A $ , $ e : \ K \rightarrow A $ , $ \mu : \ A \rightarrow A \otimes A $ , $ \epsilon : \ A \rightarrow K $ such that

i) $ ( A ,\ m ,\ e ) $ is an associative algebra with unit;

ii) $ ( A ,\ \mu ,\ \epsilon ) $ is a co-associative co-algebra with co-unit;

iii) $ e $ is a homomorphism of co-algebras;

iv) $ \epsilon $ is a homomorphism of algebras;

v) $ m $ is a homomorphism of co-algebras.

This last condition is equivalent to:

v') $ \mu $ is a homomorphism of algebras.

A grading is not assumed to be part of the definition. If there is a grading and every morphism under consideration is graded, then one speaks of a graded bi-algebra.

Let $ ( A ,\ m ,\ e ,\ \mu ,\ \epsilon ) $ be a bi-algebra over $ K $ . An antipode for the bi-algebra is a module homomorphism $ \iota : \ A \rightarrow A $ such that

vi) $ m \circ ( \iota \otimes 1 ) \circ \mu = m \circ ( 1 \otimes \iota ) \circ \mu = e \circ \epsilon $ .


A bi-algebra with antipode $ \iota $ is called a Hopf algebra. A graded Hopf algebra is a graded bi-algebra with antipode $ \iota $ which is a homomorphism of graded modules.

Given a co-algebra $ ( C ,\ \mu _{C} ,\ \epsilon _{C} ) $ and an algebra $ ( A ,\ m _{A} ,\ e _{A} ) $ , the module $ \mathop{\rm Mod}\nolimits _{K} ( C ,\ A ) $ admits a convolution product, defined as follows$$ f \star g = m _{A} \circ ( f \otimes g ) \circ \mu _{C} . $$ In terms of this convolution product conditions vi) can be stated as

vi') $ \iota \star \mathop{\rm id}\nolimits = \mathop{\rm id}\nolimits \star \iota = e \circ \epsilon $ ,


where $ \mathop{\rm id}\nolimits : \ A \rightarrow A $ is the identity morphism of the bi-algebra $ A $ .


An additional example of a Hopf algebra is the following. Let $ F _{1} ( X ; \ Y ) \dots F _{n} ( X ; \ Y ) \in K [ [ X _{1} \dots X _{n} ; \ Y _{1} \dots Y _{n} ] ] $ be a formal group. Let $ A = K [ [ X _{1} \dots X _{n} ] ] $ . Identifying $ Y _{i} $ with $ 1 \otimes X _{i} \in A \widehat \otimes A $ , the $ F _{1} \dots F _{n} $ define a (continuous) algebra morphism $ \mu : \ A \rightarrow A \widehat \otimes A $ turning $ A $ into a bi-algebra. There is an antipode making $ A $ a Hopf algebra. It is called the contravariant bi-algebra or contravariant Hopf algebra of the formal group $ F $ . Note that here the completed tensor product is used.

Hopf algebras, under the name quantum groups, and related objects have also become important in physics; in particular in connection with the quantum inverse-scattering method [a3], [a4].

References

[a1] E. Abe, "Hopf algebras" , Cambridge Univ. Press (1977) MR1857062 MR0594432 MR0321962 Zbl 0476.16008 Zbl 0236.14021
[a2] M. Hazewinkel, "Formal groups and applications" , Acad. Press (1978) MR0506881 MR0463184 Zbl 0454.14020
[a3] V.G. Drinfel'd, "Quantum groups" , Proc. Internat. Congress Mathematicians (Berkeley, 1986) , 1 , Amer. Math. Soc. (1987) pp. 798–820 Zbl 0667.16003
[a4] L.D. Faddeev, "Integrable models in (h047970148.png)-dimensional quantum field theory (Les Houches, 1982)" , Elsevier (1984) MR782509
How to Cite This Entry:
Hopf algebra. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Hopf_algebra&oldid=44252
This article was adapted from an original article by A.L. Onishchik (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article