Difference between revisions of "Co-algebra"
(Importing text file) |
(TeX partly done) |
||
Line 1: | Line 1: | ||
− | A module | + | A module $A$ over a commutative ring $k$ with two homomorphisms $\phi$ and $\epsilon$ such that the diagrams |
− | + | $$ | |
− | + | \begin{array}{ccc} A & \stackrel{\phi}{\rightarrow} & A \otimes A \\ \phi\,\downarrow & \ & \downarrow\,1 \otimes \phi \\ A \otimes A & \stackrel{\phi \otimes 1}{\longrightarrow} & A \otimes A \end{array} | |
− | + | $$ | |
and | and | ||
Revision as of 19:01, 10 April 2017
A module $A$ over a commutative ring $k$ with two homomorphisms $\phi$ and $\epsilon$ such that the diagrams $$ \begin{array}{ccc} A & \stackrel{\phi}{\rightarrow} & A \otimes A \\ \phi\,\downarrow & \ & \downarrow\,1 \otimes \phi \\ A \otimes A & \stackrel{\phi \otimes 1}{\longrightarrow} & A \otimes A \end{array} $$ and
![]() |
are commutative. In other words, a co-algebra is the dual concept (in the sense of category theory) to the concept of an associative algebra over a ring .
Co-algebras have acquired significance in connection with a number of topological applications such as, for example, the simplicial complex of a topological space, which is a co-algebra. Closely related to co-algebras are the Hopf algebras, which possess algebra and co-algebra structures simultaneously (cf. Hopf algebra).
References
[1] | S. MacLane, "Homology" , Springer (1963) |
Comments
Given a co-algebra over
. Let
be the module of
-module homomorphisms from
to
. For
define the product
by the formula
, where
is identified with
. For any two
-modules
define
by
. Then the multiplication on
can also be seen as the composite
. The element
is a unit element for this multiplication making
an associative algebra with unit, the dual algebra. In general the mapping
is not an isomorphism and there is no natural
-module homomorphism
. Thus there is no equally natural construction associating a co-algebra to an algebra over
, even when
is a field. In that case there does however exist an adjoint functor
to the functor
which associates to a co-algebra its dual algebra, i.e.
for
,
, where
and
denote, respectively, the category of
-algebras and the category of
-co-algebras, [a2]; cf. also Hopf algebra. But if
is free of finite rank over
then
is an isomorphism and the dual co-algebra can be defined.
Let be the set
. Let
and define
![]() |
Then is a co-algebra.
If and
are two co-algebras, then a morphism of co-algebras is a
-module morphism
such that
and
. A co-ideal of a co-algebra
is a
-submodule
such that
and
.
A co-module over a co-algebra
is a
-module with a
-module morphism
such that
and
the canonical isomorphism
. There are obvious notions of homomorphisms of co-modules, etc.
References
[a1] | M. Sweedler, "Hopf algebras" , Benjamin (1969) |
[a2] | E. Abe, "Hopf algebras" , Cambridge Univ. Press (1980) |
Co-algebra. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Co-algebra&oldid=40931