# Hopf algebra

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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].

## Contents

### 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

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