4.8.3 Tensor Algebra Codomain Case

Tensor Algebra Codomain Case explores how tensor algebra operates within specific codomain spaces, defining structure and transformation properties in multilinear contexts.

Tensor Algebra Codomain Case is the situation in which a tensor's multilinear map produces, upon evaluation, an element of a target space that carries not just vector addition and scalar multiplication but a full multiplicative algebra structure, so that outputs of the map can themselves be multiplied together according to the rules of that algebra. It extends the vector codomain case by requiring the codomain $W$ to be an associative (or otherwise structured) algebra over the base field, and it is the setting needed whenever a tensor's output must be composed, multiplied, or otherwise combined algebraically rather than merely added and scaled.

Formal Definition

Codomain as an Algebra

An algebra codomain case is a multilinear map

T : V_{1} \times \dots \times V_{k} \to A

where $A$ is not merely a vector space but an algebra: it is equipped with a bilinear multiplication

A \times A \to A, (a, b) \mapsto a b

that is itself bilinear over the field $F$. This structure is strictly richer than the plain vector codomain case: $A$ is simultaneously a legitimate target for multilinear maps and a space whose own elements can be combined by a second, internal operation.

Common Choices of Codomain Algebra

Typical algebra codomains encountered in tensor theory include the endomorphism algebra $\operatorname{End}(V)$ of linear operators on $V$ under composition, the exterior algebra $\Lambda(V)$ under the wedge product, the full tensor algebra $T(V) = \bigoplus_{k} V^{\otimes k}$ under concatenation, and matrix algebras $M_n(F)$ under matrix multiplication.

Consequences of an Algebra Codomain

Two Interacting Layers of Multiplication

An algebra codomain case introduces two logically distinct multiplicative structures that coexist: the multilinearity of $T$ itself, which governs how the output depends on each domain slot, and the internal multiplication of $A$, which governs how two separately computed outputs can be combined after evaluation. These two layers do not automatically interact in any particular way; any compatibility between them, such as $T$ respecting the algebra multiplication in some structured sense, must be separately verified or imposed.

Operator-Composition Codomain

When $A = \operatorname{End}(V)$, the tensor's output at each argument tuple is itself a linear operator on $V$, and the algebra's internal multiplication is operator composition. This is the setting for objects such as curvature operators or connection coefficients viewed as producing operators, where composing two such outputs (applying one operator after another) is a natural and frequently used operation distinct from any linear combination of the tensor's values.

Wedge and Concatenation Codomains

When $A = \Lambda(V)$ or $A = T(V)$, the algebra's internal multiplication is the wedge product or tensor concatenation respectively, allowing a multilinear map's outputs to be assembled into still higher-order antisymmetric or general tensors after the fact, layering the algebra's own multiplicative structure on top of the multilinearity already present in $T$.

Interaction With the Domain Structure

Algebra-Valued Contraction and Composition

Because $A$ supports internal multiplication, two multilinear maps into the same algebra codomain can be combined pointwise using that multiplication to produce a new multilinear map, defined by $(T_1 \cdot T_2)(v_1, \ldots, v_k) = T_1(v_1, \ldots, v_k), T_2(v_1, \ldots, v_k)$, provided the resulting pointwise product remains multilinear, which holds automatically when the algebra multiplication is bilinear and the domain factors are shared identically between $T_1$ and $T_2$.

Non-Commutativity Considerations

Many natural algebra codomains, such as $\operatorname{End}(V)$ or matrix algebras, are non-commutative, so the order in which two algebra-valued tensor outputs are multiplied matters; this non-commutativity must be tracked explicitly whenever tensors valued in such an algebra are combined, in contrast to the scalar or ordinary vector codomain cases, where no such ordering concern arises.

Summary of Key Points

The algebra codomain case requires the target space of a multilinear map to carry a bilinear internal multiplication, not merely vector addition and scaling.
Common examples include operator algebras under composition, exterior algebras under the wedge product, and matrix algebras under matrix multiplication.
The map's own multilinearity and the codomain's internal multiplication are independent structures unless explicit compatibility is imposed.
Multilinear maps sharing an algebra codomain can be combined pointwise using that codomain's multiplication to produce new multilinear maps.
Non-commutative algebra codomains require careful tracking of multiplication order when combining tensor outputs.