Quantum Mechanics for Mathematicians

generalizes to the case ofSU(n) for arbitraryn, where one can consider

N-fold tensor products of the defining representation ofSU(n) matrices

onCn. ForSU(n) this provides perhaps the most straightforward con-

struction of all irreducible representations of the group.

9.5 Bilinear forms and tensor products

A different sort of application of tensor products that will turn out to be im-
portant is to the description of bilinear forms, which generalize the dual space
V∗of linear forms onV. We have:

Definition(Bilinear forms).A bilinear formBon a vector spaceVover a field
k(for us,k=RorC) is a map

B: (v,v′)∈V×V→B(v,v′)∈k

that is bilinear in both entries, i.e.,

B(v+v′,v′′) =B(v,v′′) +B(v′,v′′), B(cv,v′) =cB(v,v′)

B(v,v′+v′′) =B(v,v′) +B(v,v′′), B(v,cv′) =cB(v,v′)

wherec∈k.
IfB(v′,v) =B(v,v′)the bilinear form is called symmetric, ifB(v′,v) =
−B(v,v′)it is antisymmetric.

The relation to tensor products is

Theorem 9.2.The space of bilinear forms onV is isomorphic toV∗⊗V∗.

Proof.The map

α 1 ⊗α 2 ∈V∗⊗V∗→B:B(v,v′) =α 1 (v)α 2 (v′)

provides, in a basis independent way, the isomorphism we are looking for. One
can show this is an isomorphism using a basis. Choosing a basisej ofV,
the coordinate functionsvj =e∗j provide a basis ofV∗, so thevj⊗vk will
be a basis ofV∗⊗V∗. The map above takes linear combinations of these to
bilinear forms, and is easily seen to be one-to-one and surjective for such linear
combinations.

Given a basisejofVand dual basisvjofV∗(the coordinates), the element
ofV∗⊗V∗corresponding toBcan be written as the sum
∑

j,k

Bjkvj⊗vk

This expresses the bilinear formBin terms of a matrixBwith entriesBjk,
which can be computed as
Bjk=B(ej,ek)