PlanetMath: coalgebra

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coalgebra

(Definition)

A coalgebra is a vector space over a field $\mathbb{K}$ with a $\mathbb{K}$ -linear map $\Delta\colon A \to A\otimes A$ , called the comultiplication, and a (non-zero) $\mathbb{K}$ -linear map $\varepsilon\colon A \to \mathbb{K}$ , called the counit, such that

$\displaystyle (\Delta\otimes\mathrm{id})\circ\Delta$	$\displaystyle =$	$\displaystyle (\mathrm{id}\otimes\Delta)\circ\Delta$ (coassociativity) $\displaystyle ,$
$\displaystyle (\varepsilon\otimes\mathrm{id})\circ\Delta$	$\displaystyle = \mathrm{id}=$	$\displaystyle (\mathrm{id}\otimes\varepsilon)\circ\Delta.$

In terms of commutative diagrams:

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

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

Let $\sigma\colon A\otimes A \to A\otimes A$ be the flip map $\sigma(a\otimes b) = b\otimes a$ . A coalgebra is said to be cocommutative if $\sigma\circ\Delta = \Delta$ .

Let and be two coalgebras over a field $\mathbb{K}$ . A coalgebra homomorphism is a $\mathbb{K}$ -linear map $f\colon A \to B$ such that $\Delta_B\circ f = (f\otimes f)\circ\Delta_A$ and $\varepsilon_B\circ f = \varepsilon_A$ .

Example 1 (Coalgebra of a set)
Let be a set. The free vector space $\mathbb{K}S$ , with basis given by the elements of , is a coalgebra with comultiplication $\Delta(s) = s \otimes s$ and counit $\varepsilon(s) = 1$ .

"coalgebra" is owned by mhale.

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Also defines:

comultiplication, counit, coassociative, cocommutative

Attachments:: the dual of a coalgebra is an algebra (Derivation) by mps
Newtonian coalgebra (Definition) by mps

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Cross-references: basis, homomorphism, commutative diagrams, map, field, vector space
There are 17 references to this entry.

This is version 9 of coalgebra, born on 2002-10-18, modified 2005-02-02.
Object id is 3522, canonical name is Coalgebra.
Accessed 8299 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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