# Hopf algebra

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 .

## 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) 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 ).

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 ).

The algebra $H ^{*} (G,\ K)$ in Example 4) was first considered by H. Hopf in , 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 ). 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 ). In particular, such are the cohomology algebra and the Pontryagin algebra of a connected compact Lie group over $\mathbf R$ .

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