nLab
bilinear map

Definition

For abelian groups

Definition

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

$$f:A\times B\to C$$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

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

and

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

$$(a_1,b_1)+(a_2,b_2)=(a_1+a_2,b_1+b_2).$$(a_1, b_1) + (a_2, b_2) = (a_1 + a_2 \;,\; b_1 + b_2) \,.

A group homomorphism

$$\varphi :A\times B\to C$$\phi : A \times B \to C

hence satisfies

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

and hence in particular

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

$$\varphi (a_1,b_1+b_2)=\varphi (a_1,b_1)+\varphi (0,b_2)$$\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 $A,B,C\in \mathrm{Ab}$ 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

$$f:A\times B\to A\otimes B\to C.$$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

$$f:A\times B\to C$$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

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

and

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

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

$$f:A\times B\to A{\otimes }_{R}B\to C.$$f : A \times B \to A \otimes_R B \to C \,.

Multilinear maps are again a generalisation.

For $\mathrm{\infty }$-modules

(Lurie, section 4.3.4)

See at tensor product of ∞-modules

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.

Related concepts

• binary function, bilinear map, multilinear map

• binary morphism, multimorphism

• bifunctor, Quillen bifunctor

References

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

• Jacob Lurie, Higher Algebra