group theory

# Contents

## Definition

### For abelian groups

###### Definition

For $A$, $B$ and $C$ abelian groups and $A \times B$ the cartesian product group, a bilinear map

$f : A \times B \to C$

from $A$ and $B$ to $C$ is a function of the underlying sets (that is, a binary function from $A$ and $B$ to $C$) which is a linear map – that is a group homomorphism – in each argument separately.

###### Remark

In terms of elements this means that a bilinear map $f : A \times B \to C$ is a function of sets that satisfies for all elements $a_1, a_2 \in A$ and $b_1, b_2 \in B$ the two relations

$f(a_1 + a_2, b_1) = f(a_1,b_1) + f(a_2, b_1)$

and

$f(a_1, b_1 + b_2) = f(a_1, b_1) + f(a_1, b_2) \,.$

Notice that this is not a group homomorphism out of the direct product group. The product group $A \times B$ is the group whose elements are pairs $(a,b)$ with $a \in A$ and $b \in B$, and whose group operation is

$(a_1, b_1) + (a_2, b_2) = (a_1 + a_2 \;,\; b_1 + b_2) \,.$
$\phi : A \times B \to C$

hence satisfies

$\phi( a_1+a_2, b_1 + b_2 ) = \phi(a_1,b_1) + \phi(a_2, b_2)$

and hence in particular

$\phi( a_1+a_2, b_1 ) = \phi(a_1,b_1) + \phi(a_2, 0)$
$\phi( a_1, b_1 + b_2 ) = \phi(a_1,b_1) + \phi(0, b_2)$

which is (in general) different from the behaviour of a bilinear map.

The definition of tensor product of abelian groups is precisely such that the following is an equivalent definition of bilinear map:

###### Definition

For $A, B, C \in Ab$ a function of sets $f : A \times B \to C$ is a bilinear map from $A$ and $B$ to $C$ precisely if it factors through the tensor product of abelian groups $A \otimes B$ as

$f : A \times B \to A \otimes B \to C \,.$
###### Remark

The analogous defintion for more than two arguments yields multilinear maps. There is a multicategory of abelian groups and multilinear maps between them; the bilinear maps are the binary morphisms, and the multilinear maps are the multimorphisms.

### For modules

More generally :

###### Definition

For $R$ a ring (or rig) and $A, B, C \in R$Mod being modules (say on the left, but on the right works similarly) over $R$, a bilinear map from $A$ and $B$ to $C$ is a function of the underlying sets

$f : A \times B \to C$

which is a bilinear map of the underlying abelian groups as in def. 1 and in addition such that for all $r \in R$ we have

$f(r a, b) = r f(a,b)$

and

$f(a, r b) = r f(a,b) \,.$

As before, this is equivalent to $f$ factoring through the tensor product of modules

$f : A \times B \to A \otimes_R B \to C \,.$

Multilinear maps are again a generalisation.

## Examples

• For $R = k$ a field, an $R$-module is a $k$-vector space and a $R$-bilinear map is a bilinear map out of two vector spaces.

## References

In the context of higher algebra/(∞,1)-category theory bilinear maps in an (∞,1)-category are discussed in section 4.3.4 of

Revised on June 9, 2014 09:41:10 by danel? (94.175.93.67)