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''bi-algebra, hyperalgebra''

A graded module $4$ 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;

2) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047970/h0479708.png"/> is a homomorphism of graded algebras;

3) $0$ 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 $4$ and $B$ over $K$ their tensor product $A \otimes B$ is endowed with the natural structure of a Hopf algebra. Let $A = \sum _ { n \in Z } A _ { n }$ be a Hopf algebra, where all the $A _ { n }$ are finitely-generated projective $K$-modules. Then $A ^ { * } = \sum _ { n \in Z } A _ { n } ^ { * }$, where $A _ { x } ^ { x }$ 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 $4$. An element $\pi$ of a Hopf algebra $4$ is called primitive if

\begin{equation} \delta ( x ) = x \bigotimes 1 + 1 \bigotimes x \end{equation}

The primitive elements form a graded subalgebra $P _ { A }$ in $4$ under the operation

\begin{equation} y ] = x y - ( - 1 ) ^ { p q } y x , \quad x \in A _ { p } , \quad y \in A _ { y } \end{equation}

If $4$ is connected (that is, $A _ { x } = 0$ for $n &lt; 0$, $A _ { 0 } = K$) and if $K$ is a field of characteristic 0, then the subspace $P _ { A }$ generates the algebra $4$ (with respect to multiplication) if and only if the co-multiplication is graded commutative [[#References|[2]]].

===Examples.===


1) For any graded Lie algebra $8$ (that is, a graded algebra that is a Lie [[Superalgebra|superalgebra]] under the natural $22$-grading) the universal enveloping algebra $U ( \mathfrak { g } )$ becomes a Hopf algebra if one puts

\begin{equation} \epsilon ( x ) = 0 , \quad \delta ( x ) = x \bigotimes 1 + 1 \bigotimes x , \quad x \in \mathfrak { g } \end{equation}

Here $P _ { U ( \mathfrak { g } ) } = \mathfrak { g }$. If $K$ is a field of characteristic 0, then any connected Hopf algebra $4$ generated by primitive elements is naturally isomorphic to $U ( P _ { A } )$ (see [[#References|[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 $k$.

3) The algebra of regular functions on an affine algebraic group $k$ becomes a Hopf algebra (with trivial grading) if one defines the homomorphisms $0$ and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047970/h04797051.png"/> by means of the multiplication $G \times G \rightarrow G$ and the imbedding $\{ e \} \rightarrow G$, where $E$ is the unit element of $k$ (see [[#References|[3]]]).

4) Suppose that $k$ 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 $\Delta ( \alpha ) = ( \alpha , \alpha )$, $\iota ( e ) = e$, $p ( \alpha ) = e$, $\alpha \in G$. 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 $\operatorname { mix }$, $1 , x$, $\Delta x$, $p *$ (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]]]).

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 $4$ subject to the condition $A _ { x } &lt; \infty$, $n \in Z$, over a perfect field $K$ of characteristic $D$ is described by the following theorem (see [[#References|[4]]]). The algebra $4$ splits into the tensor product of algebras with a single generator $\pi$ and the relation $x ^ { s } = 0$, where for $p = 2$, $5$ is a power of 2 or $\infty$, and for $p \neq 2$, $5$ is a power of $D$ or $\infty$ ($\infty$ for $p = 0$) if $\pi$ has even degree, and $s = 2$ if the degree of $\pi$ is odd. In particular, for $p = 0$, $4$ 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 $4$ over a field $K$ in which $x ^ { 2 } = 0$ for any element $\pi$ of odd degree and in which all elements of odd degree and all elements of even degree are decomposable, is the exterior algebra $A = \wedge P _ { A }$ (see [[#References|[2]]]). In particular, such are the cohomology algebra and the Pontryagin algebra of a connected compact Lie group over $R$.

====References====
<table><tr><td valign="top">[1]</td> <td valign="top"> H. Hopf, "Ueber die Topologie der Gruppenmannigfaltigkeiten und ihrer Verallgemeinerungen" ''Ann. of Math. (2)'' , '''42''' (1941) pp. 22–52 {{MR|}} {{ZBL|}} </td></tr><tr><td valign="top">[2]</td> <td valign="top"> J.W. Milnor, J.C. Moore, "On the structure of Hopf algebras" ''Ann. of Math. (2)'' , '''81''' : 2 (1965) pp. 211–264 {{MR|0174052}} {{ZBL|0163.28202}} </td></tr><tr><td valign="top">[3]</td> <td valign="top"> A. Borel, "Linear algebraic groups" , Benjamin (1969) {{MR|0251042}} {{ZBL|0206.49801}} {{ZBL|0186.33201}} </td></tr><tr><td valign="top">[4]</td> <td valign="top"> 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 {{MR|0051508}} {{ZBL|0052.40001}} </td></tr><tr><td valign="top">[5]</td> <td valign="top"> S. MacLane, "Homology" , Springer (1963) {{MR|}} {{ZBL|0818.18001}} {{ZBL|0328.18009}} </td></tr></table>



====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 $4$ 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) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047970/h047970123.png"/> 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 <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 $( C , \mu C , \epsilon C )$ and an algebra $( A , m _ { A } , e _ { A } )$, the module $( C , A )$ admits a convolution product, defined as follows

\begin{equation} f ^ { * } g = m _ { A } \circ ( f \otimes g ) \circ \mu _ { C } \end{equation}

In terms of this convolution product conditions vi) can be stated as

vi') $\iota * \text { id } = \text { id } * _ { \iota } = e \circ \epsilon$,

where $: A \rightarrow A$ is the identity morphism of the bi-algebra $4$.

An additional example of a Hopf algebra is the following. Let $F _ { 1 } ( X ; Y ) , \ldots , F _ { n } ( X ; Y ) \in K [ X _ { 1 } , \ldots , X _ { n } ; Y _ { 1 } , \ldots , Y _ { n } ] \}$ be a [[Formal group|formal group]]. Let $A = K [ [ X _ { 1 } , \dots , X _ { x } ] ]$. Identifying $Y$ with $1 \otimes X _ { i } \in A \otimes \sim A$, the $F _ { 1 } , \ldots , F _ { n }$ define a (continuous) algebra morphism $\mu : A \rightarrow A \otimes \cdots A$ turning $4$ into a bi-algebra. There is an antipode making $4$ a Hopf algebra. It is called the contravariant bi-algebra or contravariant Hopf algebra of the formal group $H ^ { \prime }$. 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]]].

====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 ($1 + 1$)-dimensional quantum field theory (Les Houches, 1982)" , Elsevier (1984) {{MR|782509}} {{ZBL|}} </td></tr></table>