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bilinear map

Contents

Definition

For abelian groups

Definition

For AA, BB and CC abelian groups and A×BA \times B the cartesian product group, a bilinear map

f:A×BC f : A \times B \to C

from AA and BB to CC is a function of the underlying sets (that is, a binary function from AA and BB to CC) 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×BCf : A \times B \to C is a function of sets that satisfies for all elements a 1,a 2Aa_1, a_2 \in A and b 1,b 2Bb_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×BA \times B is the group whose elements are pairs (a,b)(a,b) with aAa \in A and bBb \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

ϕ:A×BC \phi : A \times B \to C

hence satisfies

ϕ(a 1+a 2,b 1+b 2)=ϕ(a 1,b 1)+ϕ(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

ϕ(a 1+a 2,b 1)=ϕ(a 1,b 1)+ϕ(a 2,0) \phi( a_1+a_2, b_1 ) = \phi(a_1,b_1) + \phi(a_2, 0)
ϕ(a 1,b 1+b 2)=ϕ(a 1,b 1)+ϕ(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,CAbA, B, C \in Ab a function of sets f:A×BCf : A \times B \to C is a bilinear map from AA and BB to CC precisely if it factors through the tensor product of abelian groups ABA \otimes B as

f:A×BABC. 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 RR a ring (or rig) and A,B,CRA, B, C \in RMod being modules (say on the left, but on the right works similarly) over RR, a bilinear map from AA and BB to CC is a function of the underlying sets

f:A×BC 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 rRr \in R we have

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

and

f(a,rb)=rf(a,b). f(a, r b) = r f(a,b) \,.

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

f:A×BA RBC. f : A \times B \to A \otimes_R B \to C \,.

Multilinear maps are again a generalisation.

For \infty-modules

(Lurie, section 4.3.4)

See at tensor product of ∞-modules

Examples

  • For R=kR = k a field, an RR-module is a kk-vector space and a RR-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)