On the dimension and other numerical invariants of algebras and vector products

Larissa Cadorin, Max-Albert Knus, and Markus Rost

Abstract. Tensor categorical and diagrammatic techniques used in the theory of knot invariants can be applied to compute the dimension and other numerical invariants for certain algebraic structures defined by tensor identities. These techniques are described, and applied to symmetric composition algebras and 3-vector products.

Introduction

Many algebraic structures defined on vector spaces occur only in specific dimensions, independent of the base field. One classical example is given by Hurwitz’s theorem for quadratic composition algebras, which implies that such algebras exist only in dimension 1, 2 (quadratic extensions), 4 (quaternions) and 8 (octonions). Recall that quadratic composition algebras are (not necessarily associative) algebras V with unit 1 over a field F with a nonsingular quadratic form q : V → F, which is multiplicative, i.e., q(xy) = q(x)q(y) for all x, y ∈ V. The usual proofs of Hurwitz’s theorem are based on an explicit construction of the composition law and its classification. (See for example Jacobson [Jac58], or the book [KMRT98]). Moreover over algebraically closed fields the dimension is a numerical invariant which classifies quadratic composition algebras.

Vector products are also examples of algebraic structures which occur only in specific dimensions. 2-vector products are generalizations of the classical vector product in the 3-dimensional euclidian space. It is known that they can only occur in dimension 3 and 7 (see [Eck91] or [BG67]). Let b: V × V → F be a nondegenerate, symmetric bilinear form on a vector space V. A 2-vector product is a skew-symmetric bilinear map V ×V → V, (x, y) ↦ x×y, such that b(x×y, x) = b(x×y, y) = 0 and

2000 Mathematics Subject Classification. Primary 17A75; Secondary 81T99.

Key words and phrases. tensor categories, composition algebras, vector products.

b(x × y, x × y) = det |\begin{matrix}b(x, x) & b(x, y) \\ b(y, x) & b(y, y) \end{matrix}|

for all x, y ∈ V.

Recently a new, tensor theoretical approach was given in [Ros96] and [Boo98] for the calculation of the dimension of spaces admitting 2-vector products (see also [Ros04]). A variation on the argument of [Ros96] is given in [Mey02]. The linearization of the relations satisfied by 2-vector products gives tensor identities and a universal tensor category can be constructed for these relations. A canonical ring of numerical invariants is attached to this category. One proves that the dimension d of the 2-vector product algebra generates this ring of numerical invariants and satisfies the identity d(d − 1)(d − 3)(d − 7) = 0.

Inspired by knot theory and quantum field theory (see for example [BN95], [BA02] or [Tur94]), a graphical incarnation of the universal tensor category associated with 2-vector products is used in [Boo98]. This graphical representation allows to do linear algebra and tensor calculus in a handy way. The bilinear product V × V → V, (x, y) ↦ x × y is represented as a graph connecting two horizontal lines:

x  ^{y}_{x × y}  \begin{aligned}V & ⊗ & V \\ & ↓ \\ & V \end{aligned} .

A single point on a horizontal line corresponds to one copy of V and a finite set of n points to the tensor product V^{⊗n} of the same number of copies of V. Since V^{⊗0} is identified with the base field F, the symmetric bilinear form b : V × V → F is represented by the graph

x  y  \begin{aligned}V & ⊗ & V \\ & ↓ \\ & F \end{aligned} .

Any multilinear map occurring in the tensor identities satisfied by the 2-vector product can accordingly be represented by a graph connecting points on the two horizontal lines and composition of maps corresponds to gluing of diagrams. Closed graphs (i.e., without points on the two horizontal lines) represent numerical invariants. The tensor identities defining the 2-vector product translate into relations between formal sums of graphs and the above identity for the dimensions of vector products can be obtained by pure graph computations.

In this report we describe these diagrammatic techniques for a class of algebras and apply them for two specific algebraic structures: symmetric compositions and 3-vector products.

Symmetric composition algebras are algebras (without 1) with a multiplicative, nonsingular, quadratic form q such that the associated symmetric bilinear form b(x, y) = \frac{1}{2}(q(x + y) − q(x) − q(y)) is associative:

b(xy, z) = b(x, yz) for all x, y, z ∈ V.

They arise in a natural way from degree 3 alternating algebras and their structure is known (see [EM93] or[KMRT98]). In Section 5 we prove that the ring of numerical invariants of the corresponding universal tensor category is generated by two numerical invariants: the dimension d and the norm e = q(c) of a Casimir element c ∈ V (see Section 1 for the definition of the Casimir element). These two invariants satisfy the identities

(e − (d − 2)^{2})e = 0 and (d − 2)(d − 8)(d − e) = 0

with possible values

(d, e) = (0, 0), (1, 1), (2, 0), (4, 4), (8, 0) and (8, 36).

For each pair (d, e) there exists a symmetric composition algebra with invariants (d, e); further, over an algebraically closed field the pair (d, e) determines the symmetric composition algebra up to isomorphism. In particular the appearance of the second invariant e permits to distinguish the two known types of symmetric compositions in dimension 8. The computation of the ring of numerical invariants of the universal tensor category of symmetric compositions given here is due to the third named author.

A 3-vector product is an alternating trilinear product

V × V × V → V, (x, y, z) ↦ xyz,

such that

b(xyz, xyz) = det |\begin{matrix}b(x, x) & b(x, y) & b(x, z) \\ b(y, x) & b(y, y) & b(y, z) \\ b(z, x) & b(z, y) & b(z, z) \end{matrix}|

for all x, y, z ∈ V. The structure of 3-vector products is known (or more generally r-vector products, see [Eck91] or [BG67]). In particular they only occur in dimension 4 and 8. The diagrammatic interpretation of 3-vector products given in Section 6 is new and will be part of the PhD thesis [Cad05] of the first author. More applications of tensor and diagrammatic techniques to algebraic structures, including a diagrammatic proof of Hurwitz’s theorem, will be given in the thesis.

1. Algebraic structures and tensors

Let F be a field of characteristic different from 2 and let V be a finite dimensional vector space over F. Let q : V → F be a quadratic form on V with polar form b(x, y) = \frac{1}{2}(q(x+y)−q(x)−q(y)). We assume that b is nonsingular, i.e., b induces an isomorphism h : V ⟶^{∼} V^{∗} = Hom_{F} (V, F), x ↦ b(x, −).

Let V^{⊗m} = V ⊗ ⋯ ⊗ V (m copies), V^{⊗0} = F. We identify (V^{⊗m})^{∗} with (V^{∗})^{⊗m} through the canonical map (V^{∗})^{⊗m} → (V^{⊗m})^{∗}given by

(f_{1} ⊗ ⋯ ⊗ f_{m})(v_{1} ⊗ ⋯ ⊗ v_{m}) = f_{1}(v_{1}) ⋯ f_{m}(v_{m})

for f_{i} ∈ V^{∗} and v_{i} ∈ V. The dual φ^{∗} : (V^{⊗n})^{∗} → (V^{⊗m})^{∗} of any linear map φ : V^{⊗m} → V^{⊗n} then can be viewed as a linear map φ^{∗} : (V^{∗})^{⊗n} → (V^{∗})^{⊗m}. We denote by φ^{t} : V^{⊗n} → V^{⊗m} the linear map (h^{⊗m})^{−1} ∘ φ^{∗} ∘ h^{⊗n} and call it the transpose of φ. The bilinear form b can be viewed as a contraction b : V ⊗ V → F. The fact that b is symmetric can be expressed as b = b∘τ, where τ : V ⊗V → V ⊗V is the switch τ(x ⊗ y) = y ⊗ x, x, y ∈ V. We extend b to b_{k}: V^{⊗k} ⊗ V^{⊗k} → F, k ≥ 1:

b_{k}((u_{1} ⊗ ⋯ ⊗ u_{k}) ⊗ (v_{1} ⊗ ⋯ ⊗ v_{k})) = ∏_{i=1}^{k} b(u_{i}, v_{i}).

For m ≥ k, we have isomorphisms

(1.1) Hom_{F} (V^{⊗n}, V^{⊗m}) ⟶^{∼} Hom_{F} (V^{⊗(n+k)}, V^{⊗(m−k)}), θ ↦ θ^{[−k]}

where θ^{[−k]} is the composite

\underbrace{V ⊗ ⋯ ⊗ V}_{n+k factors} ⟶^{θ⊗1_{V^{⊗k}}} \underbrace{V ⊗ ⋯ ⊗ V}_{m+k factors} ⟶^{1_{V^{⊗(m−k)}}⊗b_{k}} \underbrace{V ⊗ ⋯ ⊗ V}_{m−k factors} .

We assume that b admits an orthonormal basis (e_{1}, …, e_{d}), b(e_{i}, e_{j}) = δ_{ij}, i, j = 1, …, d, extending the base field if necessary. We fix f_{i} = b(e_{i}, −), i = 1, …, d, as a dual basis of V^{∗}, so that h : V → V^{∗} is given by e_{i} ↦ f_{i}. The map b^{t}: F → V ⊗ V is given by

(1.2)  b^{t}(1_{F}) = ∑_{i} e_{i} ⊗ e_{i}.

It follows that (b ∘ b^{t})(1_{F}) = ∑_{i} b(e_{i}, e_{i}) = d, the dimension of V.

A bilinear multiplication on V is a linear map m : V ⊗V → V, x⊗y ↦ xy. We call the pair (V, m) an algebra (over F). Let (e_{1}, …, e_{d}) be an orthonormal basis of V and let m(e_{i} ⊗ e_{j}) = ∑_{k} m_{ijk}e_{k}. It follows that m^{t}(e_{k}) = ∑_{i,j} m_{ijk}e_{i} ⊗ e_{j}. If the algebra (V, m) admits an associative (or invariant) nonsingular bilinear form, i.e.,

(1.3)  b(xy, z) = b(x, yz) for all x, y, z ∈ V ,

then

(1.4)  m^{t}(v) = ∑_{i} e_{i}v ⊗ e_{i} = ∑_{i} e_{i} ⊗ ve_{i}.

For any x ∈ V, let ℓ_{x}: V → V be the linear map y ↦ xy and let Tr(ℓ_{x}) be the trace of ℓ_{x}.

Lemma 1.5. Let (V, b, m) be an algebra with a nonsingular associative bilinear form and let (e_{1}, …, e_{d}) be an orthonormal basis. The element c = ∑_{i} e_{i}e_{i} ∈ V is uniquely determined by the property Tr(ℓ_{x}) = b(c, x). In particular it is independent of the choice of the orthonormal basis.

Proof. Uniqueness follows from the fact that b is nonsingular. Let xe_{i} = ∑_{j} x_{ij}e_{j}, so that Tr(ℓ_{x}) = ∑_{i} x_{ii}. We have

∑_{i} b(x, e_{i}e_{i}) = ∑_{i} b(xe_{i}, e_{i}) = ∑_{i,j} x_{ij}b(e_{j}, e_{i}) = ∑_{i} x_{ii}.

□

The element c is usually called a Casimir element. The dimension and the value e = b(c, c) ∈ F are numerical invariants attached to the algebra (V, b, m) (where b is associative with respect to m).

For simplicity we call elements of Hom_{F} (V^{⊗n}, V^{⊗m}), for arbitrary n and m, tensors. For any triple (V, b, m), b a nonsingular symmetric bilinear form and m a multiplication, we have an alphabet of basic tensors consisting of 1_{V}, b, m, the switch τ and the transposed b^{t}, m^{t}. Let ℭ = {c_{i} ∈ Hom_{F} (V^{⊗n_{i}}, V^{⊗m_{i}})} be a set of tensors, usually generated (through compositions and tensor products) by elements from the basic alphabet. We say that an algebra (V, b, m) is of tensor type ℭ if the c_{i} are identities for (V, b, m), i.e., if

c_{i}(x_{1} ⊗ … ⊗ x_{n_{i}}) = 0

for all i and all x_{k} ∈ V.

Example 1.6. The vector product in ℝ^{3}, m: x⊗y ↦ x×y, satisfies the identities

(1.7)  \begin{aligned}x × y + y × x & = & 0 \\ x × (y × z) − ⟨x, z⟩y + ⟨x, y⟩z & = & 0 \end{aligned}

for x, y, z ∈ V, where ⟨x, y⟩ = b(x, y) is the scalar product. The tensors are m + m ∘ τ and m ∘ (1_{V} ⊗ m) − (b ⊗ 1_{V}) ∘ (1_{V} ⊗ τ) + b ⊗ 1_{V} in Hom_{F} (V^{⊗k}, V), k = 2, 3.

2. Graphs and tangles

A graph G is a finite 1-dimensional cell space. It consists of a finite set V of vertices (0-cells) and a finite set E of edges (1-cells) and is denoted by G = (V, E). Circles are admitted. Every edge e connects two vertices {v_{1}, v_{2}}, which can possibly coincide. The two endpoints determine the edge and we write e = v_{1}v_{2}. Graphs are drawn as 2-dimensional projections with the convention that braidings are symmetric:

τ =  =  .

A path P is a graph with vertices {v_{0}, …, v_{k}} and edges

{v_{0}v_{1}, v_{1}v_{2}, …, v_{k−1}v_{k}}

where the v_{i} are pairwise different, with the possible exception v_{0} = v_{k}. We say that P is a path from v_{0} to v_{k}, denote it P = v_{0}v_{1} … v_{k} and call k the length of the path. A refinement of a graph G is a graph obtained from G by dividing some of the edges, i.e., by replacing these edges with paths having the same endpoints and so that the intermediate points are not vertices of G. Given a graph G = (V, E), let G′ = (V′, E′) be the refinement obtained by adding a vertex in the “middle” of each edge. The star St(v) of a vertex v of G is the set of edges e′ ∈ E′ which are incident with v and the valency val(v) of v is the cardinality of St(v).

The set of vertices of a graph G having valency 1 is the boundary ∂G of G. We consider graphs whose vertices have valency ≤ 3. Let v be a trivalent vertex. Any bijective map Φ_{v} of {1, 2, 3} to the star St(v) of v is an ordering of the vertex v. We use the ordering to equip v with an orientation. The orientation is positive if the image of Φ_{v} is (clockwise) an even permutation and is negative otherwise. Orientations are indicated by black or white colorings:

(2.1)  (positive) ,  (negative) .

If no orientation is indicated we understand some orientation.

Let X = {x_{1}, …, x_{k}} and Y = {y_{1}, …, y_{ℓ}} be finite sets (the empty set is not excluded). A m-tangle ^{1} G : X → Y is a graph such that ∂G is the disjoint union X ⊔ Y. The sets X and Y are represented by points on two superposed horizontal lines and G connects the two sets. The set X on the upper line is the source of G and Y on the lower line is the target of G. For example

1Classical tangles do not have trivalent edges. The “m” in the definition of a m-tangle recalls that trivalent vertices represent multiplications.

y_{1} y_{2} x_{1} x_{2} y_{3} y_{4} x_{3} x_{4} y_{5} Y X

is a m-tangle from X = {x_{1}, …, x_{4}} to Y = {y_{1}, …, y_{5}}. For any X there is an obvious identity m-tangle I_{X}: X → X.

Exchanging source and target of G through a reflection along the target line we get the transposed m-tangle G^{t}: Y → X of G. The disjoint union G ⊔ G′: X ⊔ X′ → Y ⊔ Y′ of two m-tangles G: X → Y, G′: X′ → Y′ is a m-tangle. There is an obvious composition of mtangles

(G: Y → Z) × (H: X → Y) ↦ G ∘ H: X → Z

defined by gluing the source of H with the target of G. Two m-tangles with identical refinements are called isomorphic and the isomorphism class of a m-tangle G is denoted by [G]. Open paths are isomorphic to edges and closed ones to circles. Composition passes over to isomorphism classes and we may define the m-tangle category T_{1}: the objects of T_{1} are finite sets X and a morphism γ: X → Y ∈ Mor_{T_{1}}(X, Y) is the isomorphism class [G] of a m-tangle G: X → Y. The composition of two morphisms is induced by the corresponding composition of mtangles and [I_{X}] is the identity element of Mor_{T_{1}}(X, X). The disjoint union ⊔ induces a tensor product □: T_{1} × T_{1} → T_{1}. The tensor product is associative and has the unit object [∅]. For pairs of objects (X, Y), the transpose induces a map [G] ↦ [G^{t}], Mor_{T_{1}}(X, Y) → Mor_{T_{1}}(Y, X) such that ([G] ∘ [H])^{t} = [H]^{t} ∘ [G]^{t}. Note that transposing changes the orientation of 3-valent vertices.

The following morphisms

\begin{aligned}I_{\{x\}} & = & & , & β & = & , & μ & = \\ τ & = & , & & β^{t} & = & , & μ^{t} & = \end{aligned}

are called basic. They constitute the alphabet of T_{1}.

Remark 2.2. Composing μ with τ we can change the orientation of the trivalent vertex of μ. We have

=

up to isomorphisms.

Proposition 2.3. The morphisms of the category T_{1} are generated up to isomorphisms through composition and disjoint union by basic morphisms.

Remark that refinements are needed in (2.3), for example to separate adjacent trivalent vertices. Moreover, the decomposition of a morphism in basic ones is not unique. In fact the alphabet is redundant. The morphism μ^{t} can be deduced from the other basic morphisms.

The next step is to enrich the category T_{1} to a category T_{2} in such a way that, for every pair of objects X and Y, the set of morphisms Mor(X, Y) has a linear structure. Let R = ℤ[\frac{1}{2}]. The objects of T_{2} are finite sets, as for T_{1}. For two finite sets X and Y, Mor_{T_{2}}(X, Y) is the free R-module generated by the set Mor_{T_{1}}(X, Y) and composition is extended by linearity: for γ = ∑_{i} a_{i}[G_{i}], a_{i} ∈ R, [G_{i}] ∈ Mor_{T_{1}}(X, Y) and ρ = ∑_{j} b_{j}[H_{j}], b_{i} ∈ R, [H_{j}] ∈ Mor_{T_{1}}(Y, Z), we have ρ∘γ = ∑_{i,j} a_{i}b_{j}[H_{j} ∘ G_{i}] ∈ Mor_{T_{2}}(X, Z). The identity of X is by definition the element 1·[I_{X}]. The tensor product □ also extends by linearity:

H □ G = ∑_{i,j} a_{i}b_{j}[H_{j} □ G_{i}]

and the extension is again associative with unit element 1·[∅]. Composition and tensor product coincide on Mor_{T_{2}}(∅, ∅) and endow Mor_{T_{2}}(∅, ∅) with the structure of a free commutative R-algebra with 1.

In addition to generators we also can impose relations: Let

Γ = {γ_{i} ∈ Mor_{T_{2}}(X_{i}, Y_{i}), i ∈ I}

be a set of morphisms of T_{2}. For each pair of objects (X, Y) of T_{2}, the set Γ generates through compositions and tensor products a submodule M(X, Y) of Mor_{T_{2}}(X, Y). We define a new category T_{Γ} by setting

Obj(T_{Γ}) = Obj(T_{2}) and Mor_{T_{Γ}}(X, Y) = Mor_{T_{2}}(X, Y)/M(X, Y).

For any ρ ∈ Mor_{T_{2}}(X, Y), let [ρ] be its class in Mor_{T_{Γ}}(X, Y). We get a projection functor P: T_{2} → T_{Γ} such that P(X) = X for X ∈ Obj(T_{2}) and P(ρ) = [ρ] for ρ any morphism. In particular we have P(γ_{i}) = 0 for each γ_{i} ∈ Γ. We say that the γ_{i} are relations defining T_{Γ} and we call T_{Γ} the m-tangle category associated with the set of relations Γ. The algebra Mor_{T_{2}}(∅, ∅) projects on E_{Γ} = Mor_{T_{Γ}}(∅, ∅). We call E_{Γ} the algebra of numerical invariants of the category T_{Γ}. To compute E_{Γ} is the main feature of the theory. We call a m-tangle category associated with some set of tensor relations a m-tangle category of type T_{3}.

Example 2.4. The simplest nontrivial element of E_{Γ} is the circle

d = β ∘ β^{t} = .

Another interesting element of E_{Γ} is the element

e = β ∘ (μ□μ) ∘ (β^{t}□β^{t}) = .

(As we shall see the orientations do not play a rôle.)

A simple example is given by Γ = {μ}, so that μ = 0 in T_{Γ} is the only relation. Then d generates E_{Γ} and E_{Γ} is isomorphic to the polynomial ring R[d].

For any of the categories T and any object X = {x_{1}, … x_{n}}, we have a contraction β_{X}: X □ X → ∅:

β_{X}({x_{1}, …, x_{n}}, {y_{1}, …, y_{n}}) = β({x_{1}, y_{1}})□ … □β({x_{n}, y_{n}}) ∈ ∅,

where {y_{1}, …, y_{n}} is another copy of X. As for vector spaces, the contraction induces functorial isomorphisms

(2.5)  Mor_{τ} (X, Y_{1}□Y_{2}) ⟶^{∼} Mor_{τ} (X□Y_{2}, Y_{1})

given by θ ↦ θ′ = (I_{Y_{1}} □ β_{Y_{2}}) ∘ (θ □ I_{Y_{2}}). For example the image of the isomorphism class of μ^{t} ∈ Mor_{τ} ({x}, {y_{1}, y_{2}}) in Mor_{τ} ({x, y_{2}}, {y_{1}}) is the isomorphism class of μ ∘ τ.

3. Representations of m-tangle categories

Let F be a field of characteristic different from 2 and let 𝔙 = (V, b, m) be an F -algebra. The bilinear form is supposed to be associative with respect to m. Let V be the category of finite dimensional vector spaces over F. Enlarging F if necessary (which will not change the tensor type of 𝔙), we may assume that V has an orthonormal basis (e_{1}, …, e_{d}). Let I_{{x}}, β, β^{t}, μ, μ^{t}, and τ be the basic generators (the alphabet) of the m-tangle categories T_{1} and T_{2} introduced in Section 2.

Theorem 3.1. (Boos, [Boo98]) There exists a unique functor

R_{𝔙}: T_{2} → V

such that (1) R_{𝔙}(∅) = F, R_{𝔙}(X) = V_{1}⊗V_{2}⊗…⊗V_{n} for X = (x_{1}, x_{2}, …, x_{n})

and V_{i} denotes a copy of V to which x_{i} is mapped,

(2) R_{𝔙}(I_{{x}}) = 1_{V}, R_{𝔙}(τ) = τ,

(3) R_{𝔙}(β) = b, R_{𝔙}(μ) = m, R_{𝔙}(γ^{t}) = R_{𝔙}(γ)^{t} and R_{𝔙}(γ □ ρ) = R_{𝔙}(γ) ⊗ R_{𝔙}(ρ) for morphisms γ and ρ in T_{2}.

Proof. It suffices to check that R_{𝔙} is uniquely defined on the category T_{1}. The extension to T_{2} follows by linearity. The functor is clearly uniquely defined on basic morphisms (the alphabet). By Proposition 2.3 an extension to T_{1} will be unique. However, since there is no unique decomposition of morphisms in basic morphisms, the existence of an extension is not obvious. In particular there is only one type of open paths up to isomorphisms, but open paths may have a priori different realizations. We follow the clever argument given in [Boo98]. We first consider the case where T_{1} has no 3-valented vertices. Connected components are paths and it suffices to check that R_{𝔙} is well defined on paths. Paths are composed of elementary morphisms of type I_{{x}}, β, β^{t} and τ. We have

(3.2)  ^{=  =}

up to isomorphism, since open paths are isomorphic. The image of the left diagram under the functor R_{𝔙} is the linear map v ↦ ∑_{i} e_{i}b(e_{i}, v) and the image of the right diagram is v ↦ ∑_{i} b(v, e_{i})e_{i}. Both expressions are equal to v since (e_{1}, …, e_{d}) is an orthonormal basis^{2}. By induction on the length of a path, this implies that R_{𝔙} is well defined on isomorphism classes of paths which do not contain braidings. In presence of braidings we have

=  or  =

in T_{1}. This corresponds in V to the fact that the bilinear form b is symmetric. Again by induction R_{𝔙} is well defined. For the general situation it is convenient to reduce to the case where the target of any morphism ρ is empty. By (2.5) we have a canonical isomorphism

(3.3)  Mor_{T_{2}}(X, Y) ⟶^{∼} Mor_{T_{2}}(X□Y, ∅)

2In a more abstract setting, the two parts of (3.2) corresponds to the two so-called “rigidity axioms”.

which maps to

Hom_{F} (V^{⊗n}, V^{⊗m}) ⟶^{∼} Hom_{F} (V^{⊗n} ⊗ V^{⊗m}, F)

in V through the functor R_{𝔙} (see (2.5)). Thus we are reduced to the case Y = ∅ in ρ. Assume that the morphism ρ: X → ∅ contains k trivalent vertices P_{1}, … P_{k}. Refining ρ if necessary, we may assume that all P_{i} are centers of morphisms μ_{i} of type μ. Replacing the negative oriented ones by (μ_{i} ∘ τ) ∘ τ we get graphs μ_{i} ∘ τ which are positively oriented and we can assume that all orientations are positive. Let \overline{ρ} be the graph obtained from ρ by deleting all μ_{i}. Let (z_{i}^{1}, z_{i}^{2}, z_{i}^{3}) be the boundary of μ_{i} (in a positive orientation) and let Z be the finite ordered set Z = (z_{1}^{1}, z_{1}^{2}, z_{1}^{3}, …, z_{k}^{1}, z_{k}^{2}, z_{k}^{3}). The graph \overline{ρ} defines uniquely a morphism \overline{ρ}: X → Z which does not contain any μ_{i} (up to isomorphism). Thus R_{𝔙} can be uniquely extended to \overline{ρ}: X → Z. Let

(3.4) \overline{μ} = ^{z_{1}  z_{3} z_{2}} =  =  .

Composing \overline{ρ} with k copies of the morphism \overline{μ} we get back ρ. Thus, to conclude, we only have to show that the images under R_{𝔙} of the two left morphisms in (3.4) define the same multilinear maps in V. This follows from the associativity of b, b(x, yz) = b(xy, z). □

Corollary 3.5. Let 𝔙 = (V, b, m) be an algebra of tensor type ℭ = {c_{i}} for tensors c_{i} expressible through the alphabet 1_{V} , m, m^{t}, b, b^{t} and τ. Let Γ = {γ_{i}} be a chosen set of graphs in T_{2} such that R_{𝔙}(γ_{i}) = c_{i} for all c_{i} ∈ ℭ and let T_{Γ} be the corresponding category of type Γ. There is a unique functor R_{Γ} : T_{Γ} → V such that R_{𝔙} = R_{Γ} ∘ P.

Corollary 3.6. A necessary condition for the existence of an algebra of tensor type, for a set of tensors ℭ = {c_{i}}, is the existence of an R-algebra homomorphism Φ: E_{Γ} → F, where Γ = {γ_{i}} is a set of relations in T_{2} such that R_{𝔙}(γ_{i}) = c_{i} for all c_{i} ∈ ℭ.

Proof. If such an algebra exists, the functor R_{Γ} induces a homomorphism E_{Γ} = Mor_{T_{Γ}}(∅, ∅) → Hom_{F} (F, F) = F. □

Since

R_{𝔙}(β ∘ β^{t})(1_{F}) = (b ∘ b^{t})(1_{F}) = ∑_{i} b(e_{i}, e_{i}) = dim_{F} V · 1_{F}

the image of the circle d under the functor R_{𝔙} is the dimension of V. (It is convenient to use the same symbol d for the circle and the dimension.)

Rotation around the vertical symmetry axis gives the following identities

(3.7)  μ ∘ β^{t} =� =� =� .

The image under R_{Γ} of μ ∘ β^{t} is the Casimir element c = ∑_{i} e_{i}e_{i}. Thus the element

e =�

of E_{Γ} is mapped under R_{Γ} to b(c, c) · 1_{F}, which we denoted also e in Section 1.

Example 3.8. Let (V, b, m) be an algebra with m = 0, as in Example 2.4. The algebra E_{Γ} is the polynomial ring R[d], where d is the circle, and R_{Γ}(d) = dim_{F} ·1_{F}. Thus there are no obstruction to the existence of such an algebra in any dimension d (which is of course obvious!).

Example 3.9. Let (ℝ^{3}, ×) be the classical vector product with the identities (1.7). With the techniques described in the next section, it can be readily shown that for the corresponding set Γ of relations,

E_{Γ} ⟶^{∼} R[d]/d(d − 1)(d − 3).

Thus a non trivial vector product satisfying the identities (1.7) can only exist in dimension 3.

4. Associative bilinear forms

It is clear from the last step of the proof of Theorem 3.1 that the associativity of the bilinear form plays an essential rôle. The associativity has a nice diagrammatic interpretation. A 120^{o} degree clockwise rotation around the center of the diagram μ describing a multiplication does not change the m-tangle. The rotation can also be viewed as deformation:

(4.1)  ↦

and the two graphs in (4.1) are isomorphic. The image under the functor R_{𝔙} of the graph on the right corresponds to the linear map

u ⊗ v ↦ ∑_{i} b(u, ve_{i})e_{i}, u, v ∈ V

which is the multiplication map u ⊗ v ↦ uv if the bilinear form b is associative. Thus the associativity of the bilinear form corresponds to the invariance of a trivalent vertex under a cyclic permutation of the edges.

If the bilinear form is not associative with respect to the multiplication, we need to consider 6 types of trivalent vertices. Taking the two orientations in account, this can be done by distinguishing one edge (which gives the output of the multiplication under the realization functor R_{𝔙}):

x  ^{y}_{xy}  .

The 6 different trivalent vertices are:

(4.2)

and we get a much more complicated alphabet of basic m-tangles. This is the case for example for quadratic compositions with unity ([Cad05]). Observe that the second m-tangle in (4.2) is obtained from the first one through a deformation of type (4.1). A similar situation occurs for 3vector products, see Section 6.

5. Symmetric compositions

A symmetric composition algebra is an algebra (V, m) with a nonsingular associative symmetric bilinear form b such that

(5.1)  q(xy) = q(x)q(y), q(x) = b(x, x) for all x, y ∈ V.

Symmetric composition algebras were considered independently by Petersson [Pet69], Okubo [Oku78] and Faulkner [Fau88].

The linearization of (5.1) is the identity

(5.2)  b(xu, yv) + b(xv, yu) = 2b(x, y)b(u, v)

for x, y, u, v ∈ V. Thus symmetric compositions are algebras of tensorial type and there is a corresponding m-tangle category T_{sc} (Recall that the bilinear form is assumed to be associative). As in Section 2, trivalent vertices are cyclically oriented. The relation in the m-tangle category T_{sc} corresponding to (5.2) is

(5.3)  ^{^{x  u}  ^{y  v}} + ^{\begin{aligned}& u & y \\ x & & & v \end{aligned}} = 2 ^{\begin{aligned}x & & y \\ & u & & v \end{aligned}}

Applying the isomorphism (3.3) with X = {u, y} and Y = {x, v} to (5.3) we get the relation

(5.4)  \begin{matrix}u  y & & u  y & & u  y \\ � & + & � & = & 2 ·� . \\ x  v & & x  v & & x  v \end{matrix}

in T_{sc}.

Remark 5.5. It is curious to compare the relation (5.4) with the so-called IHX identity in the theory of knot invariants (see [BN95], p. 428). The IHX identity reflects the Jacobi identity of Lie algebras.

Expressed in terms of the basic alphabet, the relation (5.4) is

(1□β□1) ∘ ((τ ∘ μ^{t})□(τ ∘ μ^{t})) + μ^{t} ∘ μ ∘ τ = 2τ .

Let E_{sc} = Mor_{T_{sc}}(∅, ∅) be the ring of numerical invariants of the category T_{sc}. As in Section 3 we denote

d =� , e =�

in E_{sc}.

Theorem 5.6. Let R = ℤ[\frac{1}{2}] and let J be the ideal of the polynomial ring R[d̄, ē], generated by the elements

(5.7)  p_{1} = ē(ē − (2 − d̄)^{2})
(5.8)  p_{2} = (d̄− 2)(d̄− 8)(d̄− ē).

The homomorphism Φ: R[d̄, ē] → E_{sc} = Mor_{T_{sc}}(∅, ∅) given by ϕ(d̄) = d, ϕ(ē) = e, induces a surjective homomorphism of rings

\overline{Φ}: R[d̄, ē]/J → E_{sc}.

In particular the only possible values of the pair (d, e) of invariants are

(5.9)  (d, e) = (0, 0), (1, 1), (2, 0), (4, 4), (8, 0) and (8, 36).

Remark 5.10. We do not know whether the homomorphism \overline{Φ} in Theorem 5.6 is injective. There is the isomorphism

R[d̄, ē]/J ⊗ ℚ ≃ ℚ × ℚ × ℚ[ϵ]/(ϵ^{2}) × ℚ × ℚ × ℚ

given by

d̄↦ (0, 1, 2, 4, 8, 8), ē ↦ (0, 1, ϵ, 4, 0, 36).

On the other hand, since the 6 pairs in (5.9) can be realized by a symmetric composition algebra over ℚ (see Proposition 5.33), there is a homomorphism

E_{sc} → ℚ^{6}

with

d ↦ (0, 1, 2, 4, 8, 8), e ↦ (0, 1, 0, 4, 0, 36).

It follows that the kernel of \overline{Φ}_{ℚ} is contained in the 1-dimensional radical of R[d̄, ē]/J ⊗ ℚ.

Proof. We keep the convention that if, in a plane graph, no orientation is indicated we understand that the graph does not depend on the choice of the orientation. For oriented trivalent vertices we have the identities

� =� � =�

induced by rotation along the vertical symmetry axis.

To start with a proof, first consider the following consequence of (5.4):

(5.11) � +� = 2 ·� = 2 ·� .

This and (5) show that� is invariant under a reflection.

Together with (3.7) this means that in such a graph the orientation at the two vertices is inessential. So we may forget about them and we have

(5.12)

�  =�  =�  =�  =�  .

We have by (5) and (5.11)

(5.13) � =^{(5)}� =^{(5.11)} 2 ·� −� .

Identities (5) and (5.4) show

2 ·� =^{(5)}� +� =� +� =^{(5.4)} 2 ·� = 2d ·� .

Hence we have

(5.14)  � =� = d ·�

which implies

(5.15) � = d ·� .

By (5.12) and (5.14) this equation is valid for any choice of orientation. Relation (5.11) implies similarly

� +� = 2 ·�.

By (5.15) this gives

(5.16) �  = (2 − d) ·� .

Composing both sides with themselves yields

�  = (2 − d)^{2} ·�  = (2 − d)^{2}e.

In the next computation the second equation follows from (5.4)

2 ·�  =�  +� =^{(5.4)} 2 ·�  .

We conclude that (2−d)^{2}e = e^{2} and get the first relation e(e−(2−d)^{2}) = 0.

Going on, an application of (5.4) and of (5.14) (three times) gives

� =^{(5.4)} −�  +2·� =^{(5.14)} −d^{2}·� +2d·� .

Hence

(5.17)  �  = (2 − d)d^{2}.

A further application of (5.4) gives

�  +�  = 2 ·�  .

Hence

�  = 2 ·�  −�  .

Composing this relation with itself shows

� �  = 4·� �  −4·� �  +� �  .

We compute the three terms on the right. (5.13) gives

(5.18) � =� = 2 ·� −�  = 2d − e.

Then (5.14) and the last computation gives

� �  = d ·�  = d(2d − e).

Applying (5.13) twice for the first equation, then (5.18),(5.13),(5.4) for the second equation and finally (5.15) for the third equation gives

� �  = 4 ·�  − 4 ·� � +� � �

= 4(2d − e) − 4 · (2� � −� � � )

+(2 ·� � � −� � � )

= 4(2d − e) − 4(2e − de) + (2de − d^{2}e).
= 4(2d − e) + (2 − d)(−4e + de).

So far we have computed

� � \begin{matrix}= 4(2d − e) − 4d(2d − e) + 4(2d − e) + (2 − d)(−4e + de) \\ = (2 − d)(8d − 8e + de). \end{matrix}

The right hand side is invariant under orientation reversing. Hence

(5.19)  � �  =� � = (2 − d)(8d − 8e + de).

Next note

(5.20)  � �  +� � =^{(5.4)} 2 ·�  = 2 ·� .

For the second equality note that the two graphs are the same, they are just drawn in a different way. Relations (5.17), (5.19) and (5.20) give after dividing by 2

� �  =�  = (2 − d)d^{2}.

Putting things together:

0 = (2 − d)(8d − 8e + de) − (2 − d)d^{2} = (2 − d)(8 − d)(d − e).

This is the second desired equality.

To complete the proof of Theorem 5.6 we next check that the two graphs d and e generate the ring of numerical invariants E_{sc} = Mor_{T_{sc}}(∅, ∅). We call the elements of E_{sc} closed graphs. Since they are defined up to isomorphism, we assume that closed graphs have only trivalent vertices. Observe that a closed graph is a linear combination of “real” closed graphs.

Definition 5.21. Two graphs are called equivalent, if one can be obtained from the other by a sequence of moves

� ⟷� .

Proposition 5.22. For any graph with at least one (trivalent) vertex there is an equivalent one which contains one of the portions

�  or �  .

Proof. By a cycle S in a graph Γ we mean a simple closed path in Γ, where, in contrast with Section 2, all vertices have valence 3. By length(S) we mean the number of vertices on S.

Fix some direction on S. Then at each vertex P we have an incoming edge, an outgoing edge and the third edge not belonging to S. This triple determines an orientation at the vertex P which may be different from the given orientation in P. We now divide the vertices on S into two classes: the set of vertices for which this new orientation coincides with the given orientation of the trivalent vertex and the others. If two vertices belong to the same class, we say that they have the same orientation with respect to S. In the equivalence class of a given graph we choose a graph Γ which contains a cycle S of minimal length. If length(S) = 1, the proposition is clear. Suppose length(S) ≥ 2. For an edge σ of S the two endpoints have either the same orientation or not:

S �  or �  S

In the second case we say that σ is an alternating edge. In the first case we could make a move

(5.23)  _{S �  ⟶�  S}

and find a cycle of smaller length in a graph equivalent to Γ. Since length(S) is minimal, any edge of S is alternating. Suppose that the endpoints P, Q of an edge σ of Γ \ S lie both on S. Then the union of σ with one of the arcs on S between P and Q would have smaller length that S, except when each of these arcs have length 1. But then length(S) = 2. The last two conclusions show that in case length(S) ≥ 3 there is a portion as indicated by the first of the following pictures:

S � _{⟶}� _{⟶}_{�  S}

Now the two indicated moves would again give rise to a cycle of smaller length. So we know length(S) ≤ 2 and in case length(S) = 2 the edges of S are alternating. □

Now we consider graphs modulo the relation

(5.24)  �  +�  = 0.

Proposition 5.25. The ring of closed graphs modulo the relation (5.24) is generated by the circle d and the closed graph e.

Proof. Relation (5.24) has the following consequence (see the proof of (5.14)):

(5.26)  � = 0.

Let Γ be a closed graph. We have to show that Γ is a polynomial in d and e. We may assume that Γ is connected and argue by induction on the number of vertices in Γ.

By Proposition 5.22 we have Γ = ±Γ′, where Γ′ contains one of the portions of Proposition 5.22. In the first case we may use (5.26) and reduce the number of vertices. Let us consider the second case, i.e., Γ contains a portion

η =� .

We may assume that Γ has more than two vertices, because otherwise Γ = e. Relation (5.24) gives

(5.27) �  =�  = −�  = −�  .

Iterating this relation three times yields

(5.28)  �  = −�  .

It follows that if we change the orientation of a vertex, we merely change the sign of Γ. This is clear if the vertex is near η as in (5.28). But in general we may use (5.27) to move η to the vertex, then change the orientation and move back again using (5.27). But if we don’t have to care about orientations, then it is easy to use (5.24) to produce a further cycle of length 1. So Γ = ±Γ′, where Γ′ contains a portion

� .

But this equals (2 − d) ·� , cf. (5.16). □

Corollary 5.29. The ring E_{sc} of closed graphs is generated by d and e.

Proof. The claim follows from a filtration argument, where filtration is by the number of vertices. More precisely, consider for example the case of a cycle S in the closed graph Γ containing two black vertices:

(5.30)  S �  .

Instead of relation (5.24) which leads to steps

(5.31)  _{S �  ⟶ –�  S}

to reduce the length of S, we use the relation

(5.32) �  +�  = 2 ·� .

We claim that for S with minimal length, length(S) ≤ 2. Assume that length(S) ≥ 3. Let T be any cycle in Γ, different from S, but connected with S through the two marked black vertices in (5.30). We have length(T) ≥ length(S). For each move induced by (5.32), T is replaced by a linear combination of a cycle of length length(T)−1 (corresponding

to the term� ) and a cycle of length length(T) + length(S) − 4

(corresponding to the term� ). If length(T) = length(S) we get

in any case shorter cycles. If length(T) > length(S), S is replaced by a linear combination of cycles which contains a shorter cycle. □

Theorem 5.6 follows now from Corollary 5.29. □

All values in Theorem 5.6 can be realized:

Proposition 5.33. Let F be an algebraically closed field of characteristic different from 2 and 3. For each value (0, 0), (1, 1), (2, 0), (4, 4), (8, 0) and (8, 36) of the pair (d, e), there exists up to isomorphisms exactly one symmetric composition algebra with the given value.

Proof. Symmetric composition algebras are of two types (see for example [OO81], [EM93] or [KMRT98], Chapter VIII). The first one is related to quadratic composition algebras with identity (”Hurwitz algebras”), which, by Hurwitz’ theorem, occur in dimension 1, 2, 4 and 8. Over an algebraically closed field, Hurwitz algebras are split and there is only one type up to isomorphism for each possible dimension. The bilinear form associated with the norm is not associative, but satisfies identities of the form

(5.34)  b(xy, z) = b(x, zȳ),

where x ↦ x̄ it the conjugation map. It readily follows from (5.34) that, for the new multiplication x ⋆ y = x̄ · ȳ, the bilinear form is associative. Thus we can associate to every Hurwitz algebra, a symmetric composition algebra, which is usually called a para-Hurwitz algebra. Since F is an algebraically closed field, the standard basis (1, i, j, …) of quadratic extensions, resp. quaternions or octonions, can be normalized so that i^{2} = j^{2} = … = −1 and is orthonormal. We get for the Casimir element

c = 1 + (d − 1)(−1) = (2 − d) · 1_{F} .

Thus e = b(c, c) = (d−2)^{2} and the numerical invariants for para-Hurwitz algebras are (d, e) = (1, 1), (2, 0), (4, 4) and (8, 36).

The other type of symmetric compositions is related to separable associative cubic algebras and were first considered by Okubo (see [Oku78] or [KMRT98]). Over an algebraically closed field we have three classes of separable associative cubic algebras; A = F, F^{3} and M_{3}(F). We denote the multiplication of A by (x, y) ↦ xy. Let

p(X) = X^{3} − T (x)X^{2} + S(x)X − N(x) · 1

be the generic polynomial. The form T is linear, S is quadratic and N cubic. Let A^{0} = {x ∈ A | T (x) = 0} be the vector space of trace zero elements in A. Let q = −\frac{1}{3}S and let b be the associated symmetric bilinear form. Let ω be a primitive cubic root of 1. The bilinear form b is associative with respect to the multiplication defined on A^{0} by (x, y) ↦ x ⋆ y with

x ⋆ y = \frac{1−ω}{3} xy + \frac{2+ω}{3} yx − \frac{1}{3}T (xy)

and q(x ⋆ y) = q(x)q(y) for all x ∈ A^{0} (see for example [KMRT98]). Thus (A^{0}, q, ⋆) is a symmetric composition algebra. One checks that in all three cases the Casimir element is trivial, so that symmetric composition algebras associated with cubic algebras have invariants (d, e) = (0, 0), (2, 0) and (8, 0). The two algebras in dimension 2 (and for which e = 0) are isomorphic. In contrast, we have two types of symmetric compositions in dimension 8 which are not isomorphic, since they have different e-invariant. □

Remark 5.35. In dimension 8, one class of symmetric composition algebras (“type” G_{2}) is associated with octonions, and the second (“type” A_{2}) is associated with 3 × 3 matrices. The existence of these two types is related to the existence of two non conjugate trialitarian actions of S_{3} on Spin_{8} (see [KMRT98] or [Hel78], Chap. X, Exercise E.2). Observe that relation (5.7) has two branches, one (e = 0) corresponding to type A_{2} and the other one (e = (d − 2)^{2}) to type G_{2}. The fact that the type of a symmetric algebra of dimension 8 can be easily detected through the Casimir element is new.

6. 3-Vector products

Let V be a d-dimensional vector space over a field F of characteristic zero. Let b be a nondegenerate, symmetric bilinear form on V.

An r-vector product in V is an r-linear map

P_{r} : V^{r} = V × V × ⋯ × V ⟶ V, 1 ≤ r ≤ n,

satisfying the following properties:

b(P_{r}(v_{1}, …, v_{r}), v_{i}) = 0 for v_{i} ∈ V, i = 1, …, r (6.1)

b(P_{r}(v_{1}, …, v_{r}), P_{r}(v_{1}, …, v_{r})) = det (b(v_{i}, v_{j}))_{i,j} (6.2)

It follows from (6.1) that the multilinear form

(6.3)  f(v_{1}, …, v_{r}, v_{r+1}) = b(P_{r}(v_{1}, …, v_{r}), v_{r+1})

and that the r-vector product P_{r} are alternating. Let r = 3 and denote P_{3}(v_{1}, v_{2}, v_{3}) = v_{1}v_{2}v_{3}. Relation (6.2) becomes

(6.4)  b(xyz, xyz) = det |( \begin{matrix}b(x, x) & b(x, y) & b(x, z) \\ b(y, x) & b(y, y) & b(y, z) \\ b(z, x) & b(z, y) & b(z, z) \end{matrix} )| .

We have

(6.5)  v_{π(1)}v_{π(2)}v_{π(3)} = sgn(π)v_{1}v_{2}v_{3}, v_{i} ∈ V

and

(6.6)  b(xyz, t) = −b(x, yzt) for x, y, z and t ∈ V.

Linearizing the above identity (6.4) we get:

b(uxt, vys) + b(uxs, vyt) + b(uys, vxt) + b(uyt, vxs) = (6.7)

= 4b(x, y)b(u, v)b(s, t) − 2b(x, y)b(u, s)b(v, t) − 2b(x, y)b(u, t)b(v, s)

−2b(x, u)b(y, v)b(s, t) − 2b(x, v)b(y, u)b(s, t) − 2b(x, s)b(y, t)b(u, v) −2b(x, t)b(y, s)b(u, v) + b(x, u)b(y, t)b(v, s) + b(x, v)b(u, t)b(y, s)

+b(x, u)b(y, s)b(v, t) + b(x, v)b(u, s)b(y, t) + b(x, s)b(y, u)b(t, v)
+b(x, s)b(y, v)b(t, u) + b(x, t)b(y, u)b(s, v) + b(x, t)b(y, v)b(s, u)

for all u, v, x, y, s and t ∈ V. In the terminology of Section 2, the data (V, P_{3}, b) is an algebra of tensor type, where the product m is replaced by the 3-vector product P_{3}. The tensor relations are (6.5), (6.6) and (6.7), and we shall associate to (V, P_{3}, b) a series of m-tangle categories T_{i}, i = 1, 2, 3, as in Section 2. In particular we have again basic m-tangles I_{{x}}, τ, β and β^{t}. The main difference is that 3-valented vertices corresponding to products are replaced by 4-valented vertices corresponding to 3-vector products. Since there is no associativity for the bilinear form (a cyclic permutation of the variables in the multilinear form (6.3) changes the sign), we distinguish one edge out of every 4-valented vertex, which corresponds to the product xyz:

(6.8)  μ = ^{x}  ^{z}^{y}_{xyz}  .

In a 4-valented vertex, the (clockwise) orientation of the three black edges will always be assumed to be positive, so that we do not mark 4valented vertices. Composition of m-tangles is as in Section 2, with the rule that dotted edges can only be composed with dotted ones (because of (6.10) this can always be arranged).

We next introduce the relations corresponding to a 3-vector product in the m-tangle category T_{2}: the fact that the 3-product is alternating leads to the relation

(6.9)  ^{x_{π(1)}  x_{π(3)} x_{π(2)}} = sgn(π) ^{x_{1}  x_{3} x_{2}}

for any permutation π ∈ S_{3} of three objects. Relation (6.6) leads to

(6.10)

= −  =  = −  ,

where the second m-tangle is obtained from the first through a deformation similar to the one defined in (4.1). Relation (6.9) implies that

(6.11)  = 0 and  =  = 0.

The morphisms of T_{1} consists of m-tangles generated by the alphabet I_{{x}}, τ, β, β^{t} and μ, those of T_{2} are linear combinations with coefficients in R = ℚ. (We choose here the ring of coefficients to be ℚ to avoid computation difficulties.) The linearized formula (6.7) has the following diagrammatic translation:

+  +  +  = (6.12)

= 4  − 2  − 2  − 2  − 2

− 2  − 2  +  +  +

+  +  +  +  +  .

The step T_{2} → T_{3} is as in Section 2: we define the category T_{3vec} of type T_{3} by going modulo the relations (6.9), (6.10) and (6.12).

Let V be the category of finite dimensional vector spaces over F with morphisms linear maps. As for Theorem 3.1, we have:

Theorem 6.13. Let V be a finite dimensional vector space over a field of characteristic zero. If there exists a structure (V, P_{3}, b) of 3-vector product on V, there is a unique functor R: T_{3vec} → V sending a point to V, tensor products to tensor products, τ to τ, β to the symmetric bilinear form b and μ to the 3-multiplication P_{3}.

The functor R induces a ring homomorphism

E_{3vec} = Mor_{A_{3vec}}(∅, ∅) → F

and the main aim is to compute E_{3vec} (or at least part of it). Here again a simple element of E_{3vec} is the circle

β^{t} ∘ β = .

Since R(β^{t} ∘ β) = dim_{F} V, we denote the circle also by d.

Theorem 6.14. Let

p(d̄) = d̄(d̄− 1)(d̄− 2)(d̄+ 2)(d̄− 4)(d̄+ 4)(d̄− 8) ∈ ℚ[d̄].

The map ℚ[d̄] → E_{3vec} induced by d̄↦ d induces a ℚ-algebra homomorphism

ℚ[d̄]/p(d̄) → E_{3vec}.

Proof. We verify that p(d) = 0 in E_{3vec}. Closing (6.12) on the left (and using (6.11)) gives

(6.15)  +  = (2 − d) · (2  −  −  )

and closing on the right:

(6.16)  = −(d − 1)(d − 2) =  .

Equation (6.16) yields immediately

(6.17)  =  =  =  = −d(d − 1)(d − 2).

Moreover one finds

(6.18)  = d(d − 1)^{2}(d − 2)^{2}.

We next close (6.15) with the graph on the right as follows:

(6.19)

+  = (2−d)·(2  −  −  ).

The fourth graph is zero by (6.11). The first and the second one can be redrawn and so we get the useful relation:

(6.20)  +  = −3(d − 2)  = 3d(d − 1)(d − 2)^{2} .

We now look for an equation which permits to compute one of the two graphs on the left side of (6.20). The strategy is to compose all graphs in (6.12) on the top left and on the bottom right with copies of μ. For example we get the graph

from the first graph of (6.12). This yields to

+ 3  = −12  ,

hence (with using (6.16))

3 = −(d − 1)(d − 2)(d + 2)(d − 5) .

Connecting the dotted lines we get the first graph of (6.20):

(6.21)  =  = −\frac{1}{3}d(d − 1)(d − 2)(d + 2)(d − 5) ,

so that:

(6.22)  = \frac{1}{3}d(d − 1)(d − 2)(d^{2} + 6d − 28).

The graphs in (6.18), (6.21) and (6.22) play a central rôle in the next computations, since all other computed graphs will be linear combinations of these three.

A further application of (6.12) (compose (6.12) with μ) gives

(6.23) −  +  +  = (d − 5)(d + 2)

Moving down the three points of the source, we obtain

−  +  +  = (d − 5)(d + 2)  .

Then, composing this relation with itself:

3  + 6  = (d − 5)^{2}(d + 2)^{2}
= −d(d − 1)(d − 2)(d + 2)^{2}(d − 5)^{2}.

The graph relation (6.12) can be collated in other ways with graphs of type

and ,

30 LARISSA CADORIN, MAX-ALBERT KNUS, AND MARKUS ROST

to get on the left side of (6.12) linear combinations of the diagrams

, , and ,

and on the right side linear combinations of the diagrams

, and

which are already computed. We get a system of 4 equations

+ 2  +  = −d(d − 1)(d − 2)(d + 2)(7d − 26)
− 2  +  = −6d(d − 1)(d − 2)^{2}(d − 4)
2  +  +  = d(d − 1)(d − 2)(d^{2} + 24d − 100)
3  + 6  = −d(d − 1)(d − 2)(d + 2)^{2}(d − 5)^{2}

which has the solution

= \frac{1}{3}d(d − 1)(d − 2)(d + 2)(d^{3} − 8d^{2} − 37d + 206)
= −\frac{1}{3}d(d − 1)(d − 2)(d − 4)(d + 4)(d − 8)(d + 2)
= \frac{1}{3}d(d − 1)(d − 2)(d − 4)(d + 4)(d − 8)(d + 2)
= \frac{1}{3}d(d − 1)(d − 2)(d^{4} − 6d^{3} − 29d^{2} + 168d − 44)

Two other ways of collating (6.12) yield to the two equations:

−  =  + 2  +  ,
+ 2  −  =  −  ,

which implies  = −  = 0, hence

−\frac{1}{3}d(d − 1)(d − 2)(d + 2)(d − 4)(d + 4)(d − 8) = 0

in E. □

Corollary 6.24. The possible values for the dimension of a vector space admitting a 3-vector product are 0, 1, 2, 4 and 8. In dimension 0, 1 and 2 the product is degenerate.

Proof. The existence of a 3-vector product in dimension d^{∗} > 0 implies the existence of an evaluation map E_{3vec} → F, sending d to d^{∗}. Such a map induces in turn a homomorphism ℚ[d̄]/p(d̄) → F (where Q[d̄]/p(d̄) is as in Theorem 6.14), sending again d̄ to d^{∗}. Thus d^{∗} has to be equal to 0, 1, 2, 4 or 8. In dimensions 0, 1 and 2 the product is trivial. □

Remark 6.25. The polynomial p(d̄) also admits the negative zeroes −2 and −4. We do not know if there is another relation in E_{3vec} which would exclude these values.

Acknowledgements

The first author thanks Professor B. Eckmann for suggesting to analyse 3-vector products with diagrammatic techniques. Her research is partially supported by the Swiss National Science Foundation, Grant 21-64712.01.

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Larissa Cadorin, Mathematik Departement, ETH Zentrum, CH-8092 Zürich, Switzerland

    E-mail address: cadorin@math.ethz.ch
    URL: http://www.math.ethz.ch/~cadorin

Max-Albert Knus, Mathematik Departement, ETH Zentrum, CH-8092 Zürich, Switzerland

    E-mail address: knus@math.ethz.ch
    URL: http://www.math.ethz.ch/~knus

Markus Rost, Fakultät für Mathematik Universität Bielefeld, Postfach 100131, 33501 Bielefeld, Germany

    E-mail address: rost@mathematik.uni-bielefeld.de
    URL: http://www.mathematik.uni-bielefeld.de/~rost