1.2.5 Algebraic Tensor Object Definition
The algebraic tensor object is a multilinear map that generalizes vector spaces, forming the foundation of tensor algebra through its abstract product structure.
Algebraic Tensor Object Definition is the characterization of a tensor purely as an object within an algebraic structure — an element of a tensor product space, subject to the operations of addition, scalar multiplication, and tensor multiplication that make that space into an algebra — considered apart from any interpretation involving coordinates, physical quantities, or geometric intuition. It isolates the abstract algebraic identity of a tensor from the various concrete representations and applications through which tensors are more commonly encountered.
Isolating the Algebraic Object
A tensor can be approached from several directions: as a multi-index array obeying a transformation law, as a physical quantity such as stress or curvature, or as a multilinear map satisfying a universal property. The algebraic tensor object definition sets these interpretations aside and asks a narrower question: within the tensor algebra or a relevant tensor product space, what is a tensor purely as an algebraic element, subject only to the operations that the algebraic structure provides?
Under this definition, a tensor is simply a member of the underlying set of a tensor product space — a vector space equipped with an additional multiplication operation — and its properties are determined entirely by the axioms governing that vector space and that multiplication, without reference to any external interpretation.
The Algebraic Operations Available
Addition and Scalar Multiplication
As an element of a vector space, an algebraic tensor object can be added to any other tensor of the same type, and can be multiplied by a scalar drawn from the underlying field. These operations obey the standard vector space axioms: associativity and commutativity of addition, existence of a zero tensor and additive inverses, and compatibility between scalar multiplication and the field operations.
Tensor Multiplication
Within the tensor algebra, an additional operation is available: multiplication defined by the tensor product, which combines a tensor of one rank with a tensor of another rank to produce a tensor of the combined rank. This operation is associative but, in general, not commutative, and it is this operation, together with the underlying vector space structure, that makes the collection of all tensors into an algebra in the formal, algebraic sense of the term — a vector space equipped with a bilinear multiplication.
The expression above states the associativity of tensor multiplication, one of the defining algebraic properties that qualifies the collection of tensors, together with this operation, as an algebra in the formal sense.
Distinguishing the Algebraic Object from Its Interpretations
The value of isolating the purely algebraic tensor object lies in separating questions that belong to algebra from questions that belong to application. Whether a given algebraic tensor object correctly models a particular physical quantity, or whether its components in some basis satisfy a particular transformation law, are questions about interpretation and representation. Whether the tensor algebra is associative, whether it satisfies a universal property, or how its elements decompose into homogeneous pieces of different rank, are questions that belong entirely to the algebraic object itself and can be answered without reference to any interpretation.
This separation allows theorems about tensors to be proved once, at the purely algebraic level, and then applied uniformly across every context in which tensors are used — physics, geometry, computer science, or any other field — since the algebraic properties of the tensor object do not depend on which interpretation is subsequently placed upon it.
Relationship to Other Tensor-Related Concepts
The algebraic tensor object definition sits alongside, and underlies, the more specialized notions of a tensorial quantity, which asks whether an applied quantity qualifies as tensorial by its transformation behavior, and a tensorial relation, which asks whether an equation between tensors holds invariantly across coordinate systems. Both of these notions presuppose the prior existence of a well-defined algebraic tensor object: a tensorial quantity is a physical instance of such an object, and a tensorial relation is an equation stated among such objects using the algebraic operations that the tensor algebra provides.