2.21.5 Tensor Isomorphism Structure Preservation

Tensor Isomorphism Structure Preservation is the property that a vector-space isomorphism φ : V → W does not merely match up individual vectors between V and W, but induces compatible isomorphisms on every tensor construction built from V and W, in such a way that all of the algebraic operations defining tensor structure, addition, scalar multiplication, tensor product, contraction, symmetrization, and antisymmetrization, are carried faithfully from the tensors of V to the corresponding tensors of W. An isomorphism that preserves structure in this sense is not just a bijective linear correspondence of vectors, but a correspondence that respects every operation the tensor algebra is built from, so that computing an operation before or after applying the induced map always gives the same result.

What "Structure" Means for a Tensor Space

Beyond Linear Structure

A vector-space isomorphism φ : V → W already preserves the linear structure of V, meaning it respects addition and scalar multiplication. Tensor spaces built from V, such as T^p_q(V), Sym^k(V), and Λ^k(V), carry additional structure beyond simple linearity: a multiplication-like tensor product operation, contraction maps that pair upper and lower indices, and symmetry conditions that distinguish symmetric tensors from alternating ones. Structure preservation is the requirement that the induced map on tensors respects all of this additional apparatus, not merely the underlying vector-space addition.

The Induced Map

Given φ : V → W, the induced map on simple tensors is defined by:

and extended linearly to sums of simple tensors. Structure preservation is the fact that this extension is well defined and behaves consistently across every operation available in the tensor algebra.

Preservation of the Tensor Product Operation

Compatibility Square

The defining property of the induced map is that tensoring first and then applying φ^{⊗} = φ ⊗ φ gives the same result as applying φ to each factor first and then tensoring:

This commuting square is what it means, formally, for φ to be compatible with the tensor product operation: the two paths around the square, tensor-then-map and map-then-tensor, agree at every pair of vectors.

Consequence for Rank and Simplicity

Because the induced map sends simple tensors to simple tensors and respects linear combinations, it sends a tensor of a given rank in T^k(V) to a tensor of the same rank in T^k(W), and it never collapses a genuinely rank-k tensor into a lower-rank one, since φ^{⊗k} is itself invertible with inverse (φ^{-1})^{⊗k}.

Preservation of Contraction

Contraction Commutes with the Induced Map

For a (p, q)-tensor T on V with p, q ≥ 1, contraction over a chosen upper-lower index pair produces a (p-1, q-1)-tensor C(T). Structure preservation includes the requirement that contraction commutes with the induced maps on T^p_q(V) and T^p_q(W):

so that contracting after mapping, or mapping after contracting, both land on the identical tensor in the lower-rank space; contraction is a natural operation with respect to isomorphisms, not merely a coordinate bookkeeping device.

Full Contractions as Preserved Scalars

When p = q and every index is contracted, C(T) is a scalar, and the compatibility of contraction with φ^{⊗} forces this scalar to be identical whether it is computed from T in V or from the corresponding tensor in W; full contraction is an invariant that the isomorphism carries across unchanged, rather than merely transformed.

Preservation of Symmetric and Alternating Structure

Symmetric Tensors Map to Symmetric Tensors

The induced map φ^{⊗k} commutes with the action of the permutation group on T^k(V) and T^k(W), since permuting tensor factors before or after applying φ to each factor gives the same rearranged tensor. As a direct consequence, φ^{⊗k} restricts to a well-defined isomorphism between Sym^k(V) and Sym^k(W), sending a symmetric tensor to a symmetric tensor rather than to some tensor lacking the symmetry property.

Alternating Tensors Map to Alternating Tensors

The same permutation-commuting property shows that φ^{⊗k} restricts to an isomorphism between Λ^k(V) and Λ^k(W), sending an alternating tensor, one that changes sign under transposition of any two arguments, to an alternating tensor in Λ^k(W); the sign-change condition is preserved exactly because φ^{⊗k} commutes with every permutation of tensor slots, including transpositions.

Why Structure Preservation Is the Correct Notion of Tensor-Space Isomorphism

Beyond a Mere Dimension Match

Two tensor spaces can share the same dimension without there being any operation-respecting correspondence between them, as noted when comparing symmetric and alternating tensor spaces of matching dimension. Structure preservation is what distinguishes an isomorphism that merely matches vectors, or merely matches dimension, from one that faithfully carries the entire algebraic apparatus, tensor product, contraction, symmetrization, antisymmetrization, from one space to the other.

Structure Preservation as a Functorial Property

Because the induced maps φ ↦ φ^{⊗k}, φ ↦ φ^*, φ ↦ φ|_{Sym^k}, and φ ↦ φ|_{Λ^k} are each built by the same uniform recipe and each respect composition, (ψ ∘ φ)^{⊗k} = ψ^{⊗k} ∘ φ^{⊗k}, and identity, (id_V)^{⊗k} = id_{T^k(V)}, structure preservation is not a coincidental feature of any one isomorphism but a systematic consequence of how tensor constructions are built functorially out of the base vector space.