14.2.5 Tensor Product Map Composition Area

Explore how tensor product maps compose in algebra, revealing their structure and role in tensor algebra operations.

Tensor Product Map Composition Area is the branch of study concerned with how tensor products of linear maps interact with the operation of composition, establishing the identities that let a chain of tensor products of maps be composed factor by factor rather than as opaque maps on the full tensor product.

The Core Composition Identity

Statement of the Identity

For linear maps

f : V_{1} \to W_{1}

g : V_{2} \to W_{2}

f^{'} : W_{1} \to U_{1}

g^{'} : W_{2} \to U_{2}

the composition area establishes that

(f^{'} \otimes g^{'}) \circ (f \otimes g) = (f^{'} \circ f) \otimes (g^{'} \circ g),

reducing the composite of two tensor products of maps to a single tensor product of two composites, one composite per factor.

Proof by Reduction to Elementary Tensors

The identity is established by checking both sides on an elementary tensor v tensor w, where the left side gives f prime of f of v tensor g prime of g of w and the right side gives the same expression directly from evaluating the composite maps, so the two sides agree on every elementary tensor. Since elementary tensors span V1 tensor V2, and both sides are linear maps, agreement on elementary tensors forces agreement everywhere.

Consequences of the Composition Identity

Reduction of Chains

The composition identity extends inductively to any finite chain of tensor products of maps, allowing a long composite

(f_{k} \otimes g_{k}) \circ \dots \circ (f_{1} \otimes g_{1})

to be rewritten as the single tensor product

(f_{k} \circ \dots \circ f_{1}) \otimes (g_{k} \circ \dots \circ g_{1}),

so that computing a long chain of tensor product maps reduces to composing each factor's own chain separately and tensoring only once at the end.

Compatibility with Identities

Because tensoring identity maps produces an identity map on the tensor product, the composition identity together with this fact shows that the operation sending a pair of maps to their tensor product respects both composition and identities, which is exactly the requirement for this operation to define a functor of two variables on the category of vector spaces.

Invertibility Under Composition

If f and g are each invertible, applying the composition identity to f and its inverse, and to g and its inverse, shows directly that

(f \otimes g) \circ (f^{-1} \otimes g^{-1}) = {id}_{W_{1}} \otimes {id}_{W_{2}} = {id}_{W_{1} \otimes W_{2}},

confirming that f tensor g is invertible with inverse f inverse tensor g inverse, without needing any separate argument about the tensor product map beyond the composition identity already established.

Composition with Partial Tensoring

Factorization Through Identity Extensions

The general tensor product f tensor g can itself be factored, via the composition identity, into two simpler steps that each tensor only one map with an identity:

f \otimes g = (f \otimes {id}_{W_{2}}) \circ ({id}_{V_{1}} \otimes g),

obtained from the composition identity by writing f as f composed with the identity and g as the identity composed with g. This factorization shows that any tensor product of maps can be built by first acting on one factor and then on the other, in either order, since the two orders agree by the commuting-actions property established for the composition identity.

Order Independence

Reversing the factorization order, so that g is applied first and f afterward, gives an equal result,

({id}_{W_{1}} \otimes g) \circ (f \otimes {id}_{V_{2}}) = f \otimes g,

which follows again from the composition identity applied to this alternate factorization, confirming that the two single-factor extensions of f and g commute as operations on the tensor product.