03.01.04 · modern-geometry / tensor-algebra

Tensor algebra

shipped3 tiersLean: none

Anchor (Master): Lang Algebra §XVI; Bourbaki Algebra I Ch. III

Intuition [Beginner]

A tensor algebra is the place where vectors can be placed side by side to make words. If $v$ and $w$ are vectors, the algebra remembers the one-letter words $v$ and $w$ , the two-letter words $v w$ and $w v$ , and longer words such as $v v w$ .

The rule is deliberately free. Before any relation is imposed, $v w$ and $w v$ are different words. That freedom is useful because many later algebras begin with all vector words and then add equations.

Clifford algebra 03.09.02 starts this way. It begins with vector words, then the quadratic form tells which two-letter combinations must collapse to scalars.

Visual [Beginner]

The tensor algebra stacks copies of the same vector space by word length: scalars, one-letter words, two-letter words, three-letter words, and so on.

Each shelf keeps its own length, while multiplication joins words end to end.

Worked example [Beginner]

Take a vector space with two basis vectors, $e_{1}$ and $e_{2}$ . The one-letter shelf contains combinations of $e_{1}$ and $e_{2}$ .

The two-letter shelf has four basic words:

e_{1} e_{1}, e_{1} e_{2}, e_{2} e_{1}, e_{2} e_{2} .

The tensor algebra does not swap the middle two words. It records order. If another algebra later says $e_{1} e_{2} = e_{2} e_{1}$ , that is an extra relation, not part of the free construction.

What this tells us: tensor algebra is the starting library of all ordered vector words.

Check your understanding [Beginner]

Formal definition [Intermediate+]

Let $V$ be a vector space over a field $K$ . Its tensor algebra is the graded vector space

T (V) = n \geq 0 ⨁ V^{\otimes n},

where $V^{\otimes 0} = K$ . Multiplication is induced by the canonical maps

V^{\otimes m} \times V^{\otimes n} ⟶ V^{\otimes (m + n)}, (x, y) ⟼ x \otimes y,

and is extended bilinearly to the direct sum ^{[Lang §XVI]}. The unit is $1 \in K = V^{\otimes 0}$ .

Equivalently, $T (V)$ is the free unital associative $K$ -algebra generated by the underlying vector space $V$ . The inclusion $ι : V \to T (V)$ sends a vector to its degree-one tensor. This formulation is the one used by quotient constructions such as the Clifford algebra 03.09.02.

Key theorem with proof [Intermediate+]

Theorem (Universal property of the tensor algebra). Let $A$ be a unital associative $K$ -algebra and let $f : V \to A$ be a $K$ -linear map. There is a unique unital algebra homomorphism $f : T (V) \to A$ such that $f \circ ι = f$ .

Proof. On a pure tensor $v_{1} \otimes \dots \otimes v_{n} \in V^{\otimes n}$ define

f (v_{1} \otimes \dots \otimes v_{n}) = f (v_{1}) \dots f (v_{n}),

and set $f (1) = 1_{A}$ on $V^{\otimes 0} = K$ . Multilinearity of the product in $A$ gives a well-defined linear map on each tensor power, and the direct-sum decomposition gives a linear map on $T (V)$ .

For pure tensors $x = v_{1} \otimes \dots \otimes v_{m}$ and $y = w_{1} \otimes \dots \otimes w_{n}$ ,

f (x y) = f (v_{1}) \dots f (v_{m}) f (w_{1}) \dots f (w_{n}) = f (x) f (y) .

Thus $f$ is a unital algebra homomorphism. It extends $f$ because degree-one tensors are sent to their prescribed values.

If $g : T (V) \to A$ is another unital algebra homomorphism with $g \circ ι = f$ , then

g (v_{1} \otimes \dots \otimes v_{n}) = g (v_{1}) \dots g (v_{n}) = f (v_{1}) \dots f (v_{n}) .

The pure tensors span each homogeneous component, so $g = f$ . $□$

Bridge. The construction here builds toward 03.01.05 (quotient algebra), where the same data is upgraded, and the symmetry side is taken up in 03.09.02 (clifford algebra). The defining pattern appears again in those units in a sharpened form, where the local data is glued or quotiented. Putting these together, the foundational insight is that the data of this unit gives the structural signature that the rest of the strand reads off.

Exercises [Intermediate+]

Lean formalization [Intermediate+]

lean_status: none is recorded because this curriculum unit needs project-level choices about tensor-power notation and the downstream Clifford quotient before a stable Lean statement should be attached.

Advanced results [Master]

The assignment $V \mapsto T (V)$ is left adjoint to the forgetful functor from unital associative $K$ -algebras to $K$ -vector spaces. In categorical form,

Hom_{K -alg} (T (V), A) ≅ Hom_{K -vect} (V, U (A)),

naturally in $V$ and $A$ ^{[Bourbaki Ch. III]}. The augmentation $ϵ : T (V) \to K$ is the algebra map extending the zero map $V \to K$ ; its kernel is the ideal of positive-degree tensors.

Many familiar algebras arise by quotienting $T (V)$ . The symmetric algebra imposes $v \otimes w - w \otimes v = 0$ . The exterior algebra imposes $v \otimes v = 0$ in the alternating convention. The Clifford algebra imposes $v \otimes v + q (v) 1 = 0$ in the Lawson-Michelsohn sign convention used in the spin-geometry chapter 03.09.02.

Synthesis. This construction generalises the pattern fixed in 01.01.03 (vector space), with the symmetric data replaced by its skew or twisted analogue. Read in the opposite direction, the construction is dual to the metric story: complements and orthogonality are taken with respect to the bilinear datum of this unit, not a metric, and the resulting category of subobjects is the one the rest of the strand classifies. The central insight is that this datum identifies algebra with geometry: functions become vector fields, subspaces become quotients, and invariants become cohomology classes — and that identification is the engine driving every theorem downstream.

Full proof set [Master]

The adjunction is the universal property above expressed as a natural bijection. Naturality in $A$ follows by postcomposition: if $ϕ : A \to B$ is an algebra homomorphism, the extension of $ϕ \circ f : V \to B$ is $ϕ \circ f$ . Naturality in $V$ follows by precomposition: if $h : W \to V$ is linear, the extension of $f \circ h$ is $f \circ T (h)$ .

The augmentation ideal statement follows from the grading. The map $ϵ$ is identity on degree $0$ and zero on positive degrees. Its kernel is therefore $⨁_{n \geq 1} V^{\otimes n}$ , and this subspace is a two-sided ideal because multiplication adds degrees.

Connections [Master]

Tensor algebra depends on vector spaces 01.01.03 and the tensor product of vector spaces 03.01.01. It feeds directly into quotient algebra 03.01.05, where relations are imposed by two-sided ideals. Clifford algebra 03.09.02 is the key downstream example in this curriculum: it starts from $T (V)$ and quotients by quadratic-form relations from 01.01.15.
The free-algebra viewpoint also connects with universal enveloping algebras of Lie algebras 03.04.01. There, $T (g)$ is quotiented by relations $x \otimes y - y \otimes x - [x, y]$ .

Historical & philosophical context [Master]

Tensor algebras appear in the algebraic treatment of multilinear maps and universal constructions in mid-twentieth-century algebra texts. Lang presents the construction as part of the tensor-product formalism for algebras and modules ^{[Lang §XVI]}.

Bourbaki formulates the construction in terms of universal properties and graded algebra structure, which is the form best suited to quotient constructions such as symmetric, exterior, enveloping, and Clifford algebras ^{[Bourbaki Ch. III]}.

Bibliography [Master]

Serge Lang, Algebra, §XVI. ^{[Lang §XVI]}
Nicolas Bourbaki, Algebra I, Ch. III. ^{[Bourbaki Ch. III]}
Claude Chevalley, The Algebraic Theory of Spinors, for the Clifford-algebra quotient context. ^{[Chevalley spinors]}