PlanetMath: Hopf algebra

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Hopf algebra

(Definition)

A Hopf algebra is a bialgebra over a field $\mathbb{K}$ with a $\mathbb{K}$ -linear map $S : A \to A$ , called the antipode, such that

$\displaystyle m\circ(S\otimes\mathrm{id})\circ\Delta = \eta\circ\varepsilon = m\circ(\mathrm{id}\otimes S)\circ\Delta,$

(1)

where $m : A\otimes A \to A$ is the multiplication map $m(a\otimes b) = ab$ and $\eta : \mathbb{K}\to A$ is the unit map $\eta(k) = k\mathord{\mathrm{1\!\!\!\:I}}$ .

In terms of a commutative diagram:

$\displaystyle \begin{xy} \xymatrix @R=20pt@C=70pt{ & *+<10pt>\txt{$A$} \ar_{\De... ...& & *+<10pt>\txt{$A\otimes A$} \ar^{m}[dl] \ & *+<10pt>\txt{$A$} & } \end{xy}$

The category of commutative Hopf algebras is anti-equivalent to the category of affine group schemes. The prime spectrum of a commutative Hopf algebra is an affine group scheme of multiplicative units. And going in the opposite direction, the algebra of natural transformations from an affine group scheme to its affine 1-space is a commutative Hopf algebra, with coalgebra structure given by dualising the group structure of the affine group scheme. Further, a commutative Hopf algebra is a cogroup object in the category of commutative algebras.

Example 1 (Algebra of functions on a finite group)
Let be the algebra of complex-valued functions on a finite group and identify $C(G\times G)$ with $A \otimes A$ . Then, is a Hopf algebra with comultiplication $(\Delta(f))(x,y) = f(xy)$ , counit $\varepsilon(f) = f(e)$ , and antipode $(S(f))(x) = f(x^{-1})$ .

Example 2 (Group algebra of a finite group)
Let $A = \mathbb{C}G$ be the complex group algebra of a finite group . Then, is a Hopf algebra with comultiplication $\Delta(g) = g \otimes g$ , counit $\varepsilon(g) = 1$ , and antipode $S(g) = g^{-1}$ .

The above two examples are dual to one another. Define a bilinear form $C(G) \otimes \mathbb{C}G \to \mathbb{C}$ by $\langle f,x \rangle = f(x)$ . Then,

$\displaystyle \langle fg, x \rangle$	$\displaystyle =$	$\displaystyle \langle f \otimes g, \Delta(x) \rangle,$
$\displaystyle \langle 1, x \rangle$	$\displaystyle =$	$\displaystyle \varepsilon(x),$
$\displaystyle \langle \Delta(f), x \otimes y \rangle$	$\displaystyle =$	$\displaystyle \langle f, xy \rangle,$
$\displaystyle \varepsilon(f)$	$\displaystyle =$	$\displaystyle \langle f, e \rangle,$
$\displaystyle \langle S(f), x \rangle$	$\displaystyle =$	$\displaystyle \langle f, S(x) \rangle,$
$\displaystyle \langle f^*, x \rangle$	$\displaystyle =$	$\displaystyle \overline{\langle f, S(x)^* \rangle}.$

Example 3 (Polynomial functions on a Lie group)
Let $A = \mathrm{Poly}(G)$ be the algebra of complex-valued polynomial functions on a complex Lie group and identify $\mathrm{Poly}(G\times G)$ with $A \otimes A$ . Then, is a Hopf algebra with comultiplication $(\Delta(f))(x,y) = f(xy)$ , counit $\varepsilon(f) = f(e)$ , and antipode $(S(f))(x) = f(x^{-1})$ .

Example 4 (Universal enveloping algebra of a Lie algebra)
Let $A = \mathcal{U}(\mathfrak{g})$ be the universal enveloping algebra of a complex Lie algebra $\mathfrak{g}$ . Then, is a Hopf algebra with comultiplication $\Delta(X) = X\otimes1+1\otimes X$ , counit $\varepsilon(X) = 0$ , and antipode .

The above two examples are dual to one another (if $\mathfrak{g}$ is the Lie algebra of ). Define a bilinear form $\mathrm{Poly}(G) \otimes \mathcal{U}(\mathfrak{g}) \to \mathbb{C}$ by $\langle f,X \rangle = \frac{\mathrm{d}}{\mathrm{d}t}\big\vert _{t=0} f(\exp(tX))$ .

"Hopf algebra" is owned by mhale.

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Also defines:

antipode

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Cross-references: universal enveloping algebra, Lie algebra, polynomial functions, Lie group, bilinear form, group algebra, complex, counit, comultiplication, finite group, functions, algebras, object, group, structure, coalgebra, natural transformations, algebra, opposite, group scheme of multiplicative units, prime spectrum, group schemes, commutative, category, commutative diagram, unit, multiplication, map, field, bialgebra
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This is version 9 of Hopf algebra, born on 2002-10-18, modified 2005-07-12.
Object id is 3524, canonical name is HopfAlgebra.
Accessed 8946 times total.

Classification:

AMS MSC:

16W30 (Associative rings and algebras :: Rings and algebras with additional structure :: Coalgebras, bialgebras, Hopf algebras ; rings, modules, etc. on which these act)

Pending Errata and Addenda

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