Scalar tensor automorphisms and affine group reconstruction

Written by GPT-6.1 Sol (OpenAI), October 2026. Public domain (CC0).

A tensor automorphism is a compatible family of matrices, one on every fibre. To recover its parameter space, we must allow matrices over every commutative scalar algebra, including algebras with nilpotents. Field-valued matrices alone can miss an entire group.

Let \(k\) be any field, \(A\) an essentially small \(k\)-linear abelian rigid symmetric monoidal category, and \(F:A\to\operatorname{Vect}^{\rm fd}_k\) an exact faithful symmetric strong monoidal functor. Rigid means that both left and right duals exist. Fix its tensor and unit comparisons \(J,j_0\). All categories and sums are small in suitable universes. The scalar algebras below are unital commutative \(k\)-algebras; the zero algebra is allowed.

Coend coalgebras and reconstruction from finite fibres, Sections 1–6 constructs the coefficient coalgebra \(H\), its coactions, and the full equivalence with finite right comodules. Monoidal fibres, Sections 1–6 constructs its product and antipode, proves that the symmetry makes \(H\) commutative, and upgrades that equivalence to a symmetric strong monoidal equivalence. We use these actual constructions, not a new choice of fibre functor.

Write \(h_{ij}=[e_i^*\otimes e_j]_a\) for a basis of \(F(a)\). The conventions are

The coefficient relation is \([\psi\otimes F(f)v]_b=[\psi F(f)\otimes v]_a\). These coefficients span \(H\) as a vector space. The linked reconstruction proof also identifies this \(H\) with the coefficient Hopf algebra of all its finite comodules. [EGNO], Example 5.4.4(i), gives classical affine-group context.

1. Natural operators after arbitrary scalar extension

For an algebra \(R\), set \(F_R(a)=F(a)\otimes_k R\), with its induced strong monoidal comparisons over \(R\). A natural operator on \(F_R\) means a family of \(R\)-linear endomorphisms, natural for the arrows of \(A\). It need not be obtained by tensoring a \(k\)-valued operator with \(R\).

Proposition 1.1. There is a natural bijection

Proof. A family \(\eta\) defines a linear functional on each coefficient summand by \(\phi\otimes v\mapsto(\phi\otimes1)\eta_a(v\otimes1)\). Naturality for \(f:a\to b\) says exactly that these functionals annihilate the displayed coefficient relation. They therefore descend to a unique \(k\)-linear \(\lambda:H\to R\).

Conversely, the formula in (1.1), extended \(R\)-linearly, is basis independent: it is the composite of the canonical coaction and the contraction with \(\lambda\). The coefficient relation makes it natural for every \(f\). Evaluating by every \(\phi\) proves that the two constructions are inverse. Finite dimensionality of each individual fibre justifies its matrix expansion; no bound on all fibre dimensions or dimension of \(H\) is needed.

For \(q:R\to R'\), tensoring each matrix entry with \(R'\) replaces \(\lambda\) by \(q\lambda\). Thus the bijection is natural in the scalar algebra as well as in the coefficient objects. \(\square\)

Lemma 1.2. Composition of operators corresponds to convolution, in the order

Proof. Applying the two matrices successively to \(e_j\) gives the \(e_i\)-coefficient \(\sum_l\lambda(h_{il})\mu(h_{lj})\), which is (1.2) by (0.1). The identity operator corresponds to \(h\mapsto\epsilon(h)1_R\). Associativity and its unit also follow directly by expanding coassociativity and the counit equations. \(\square\)

2. Tensor compatibility gives the representing algebra

An operator \(\eta\) is monoidal when it preserves the specified tensor comparisons and unit. In particular its component on \(F_R(\mathbf1)\) is identity.

Theorem 2.1. Under (1.1), monoidal natural operators are exactly the unital algebra maps \(H\to R\). Every such operator is invertible. Consequently

These identifications are natural group isomorphisms in \(R\).

Proof: tensor and unit. For coefficients \(x=[\phi\otimes v]_a\) and \(y=[\psi\otimes w]_b\), the product construction in Monoidal fibres is

Evaluate the tensor-compatibility equation for \(\eta\) on \(v,w\), and then with \((\phi\otimes\psi)J^{-1}\). It becomes \(\lambda(xy)=\lambda(x)\lambda(y)\). Coefficients span \(H\), so bilinearity gives this identity for arbitrary \(x,y\). The unit comparison gives \(\lambda(1_H)=1_R\). The same evaluations in reverse prove the converse: elementary tensors and their dual evaluations detect equality of the two \(R\)-linear tensor maps.

Proof: inverses and groups. Since \(S\) reverses multiplication and \(R\) is commutative, \(\lambda S\) is again a unital algebra map. The two antipode equations give \(\lambda*(\lambda S)=\epsilon_R=(\lambda S)*\lambda\). Hence both inverse compositions are identity by Lemma 1.2.

Convolution of two algebra maps is an algebra map: expand \(\Delta(xy)=\Delta(x)\Delta(y)\) and commute the values in \(R\) to obtain multiplicativity; the unit is immediate. The counit is an algebra map. Scalar extension preserves all these formulas. This proves every assertion, including that monoidal endomorphisms already are automorphisms. \(\square\)

Here \(\epsilon_R(h)=\epsilon(h)1_R\). For the zero algebra every free module and every matrix family is zero, and the unique maps in (2.1) give the one-element group. No exceptional choice of a nonzero scalar algebra is hidden.

3. What the affine group scheme means

For a commutative \(k\)-algebra \(B\), write \(\operatorname{Spec}(B)\) for its affine functor \(R\mapsto\operatorname{Alg}_k(B,R)\). A morphism of these affine functors means a natural transformation on all scalar algebras. This is the functor-of-points description of affine schemes; only this description is needed here.

Lemma 3.1. Every natural transformation \(\operatorname{Spec}(B)\to\operatorname{Spec}(C)\) is precomposition by a unique algebra map \(q:C\to B\).

Proof. Evaluate the transformation at \(R=B\) and at \(\operatorname{id}_B\), obtaining \(q\). Naturality for any \(\lambda:B\to R\) says that its value at \(\lambda\) is \(\lambda q\). This proves existence, uniqueness and the converse. Thus it is essential that the functors are tested on all algebras, including \(B\) itself. \(\square\)

The universal property of the ordinary tensor product of commutative algebras gives

Indeed a map from \(B\otimes C\) is determined by its restrictions \(b\mapsto b\otimes1\) and \(c\mapsto1\otimes c\). Conversely a pair \(f,g\) defines \(b\otimes c\mapsto f(b)g(c)\); the commuting values in \(R\) make it an algebra map, and the two constructions are inverse.

Corollary 3.2. The functor in (2.1) is the affine group scheme \(\operatorname{Spec}(H)\), with multiplication represented by \(\Delta\), identity by \(\epsilon\), and inverse by \(S\).

Proof. Formula (3.1) turns \(\Delta:H\to H\otimes H\) into a natural multiplication. The formulas in (2.1) identify it with composition of the original tensor automorphisms. Alternatively, the Hopf identities become the group identities by Lemma 3.1. Because \(H\) is commutative, its anti-algebra antipode is an algebra map and does represent a morphism of affine functors.

More generally a natural group homomorphism \(\operatorname{Spec}(B)\to\operatorname{Spec}(C)\), for commutative Hopf algebras \(B,C\), corresponds exactly to an algebra map \(q:C\to B\) preserving coproduct and counit. Evaluate the multiplication equation at \(R=B\otimes B\) on the pair of universal maps \(b\mapsto b\otimes1,1\otimes b\). It gives \(\Delta_B q=(q\otimes q)\Delta_C\). Evaluate the identity equation at \(R=k\) to get \(\epsilon_Bq=\epsilon_C\). Conversely these equalities prove the group and unit equations for every pair of points by composition with their maps into \(R\). The antipode is then preserved by uniqueness of a convolution inverse, proved in Monoidal fibres, Section 4. \(\square\)

This identifies the actual representing object and its maps. No assertion that points over \(k\), or over algebraic closures, suffice has been used.

4. Representations are exactly the coefficient comodules

Let \(G=\operatorname{Spec}(H)\). A finite-dimensional representation of \(G\) is a vector space \(V\) and a natural group homomorphism \(G(R)\to\operatorname{GL}_R(V\otimes R)\) on all scalar algebras. A morphism is a \(k\)-linear map intertwining those actions for every \(R\).

Theorem 4.1. Such representations and finite right \(H\)-comodules are the same category, including their symmetric tensor structures.

Proof: a comodule gives an action. Put \[ g_\lambda(v\otimes1) =\sum v_{(0)}\otimes\lambda(v_{(1)}). \tag{4.1} \] The counit gives identity at \(\epsilon\); coassociativity gives \(g_\lambda g_\mu=g_{\lambda*\mu}\), with the same order as (1.2). The antipode supplies the inverse. The formula is natural in \(R\). Colinear maps intertwine all these actions.

Proof: an action gives a comodule. Use the point \(\operatorname{id}_H\in G(H)\) and define \(\rho(v)=g_{\operatorname{id}_H}(v\otimes1)\in V\otimes H\). Naturality for an arbitrary \(\lambda:H\to R\) recovers exactly (4.1). At the two points \(i_1,i_2:H\to H\otimes H\), their convolution is the point \(\Delta\). Naturality and the group law therefore say that the coefficient matrix \((h_{ij})\) of \(\rho\) satisfies \(\Delta h_{ij}=\sum_l h_{il}\otimes h_{lj}\). That is coassociativity. Naturality for \(\epsilon:H\to k\) and the identity action gives the counit. These computations do not assume the matrix coefficients are algebraically independent.

For a map of representations, the intertwining equation at \(R=H,\lambda=\operatorname{id}_H\) is exactly colinearity. The two constructions on objects and arrows are visibly inverse. For tensor products, (4.1) multiplies the two coefficients, exactly the tensor-comodule rule in Monoidal fibres, Section 2. The unit is the counit action, and commutativity of \(H\) makes the ordinary flip compatible with both structures. This proves the asserted symmetric tensor identification. \(\square\)

Combining this theorem with the two linked reconstruction lessons gives an exact faithful symmetric strong monoidal equivalence

Under it the original \(F\) is the underlying-vector-space functor, with its specified comparisons. Neither a finite number of simple objects nor semisimplicity nor an algebraic closure of \(k\) is assumed. This is a neutral reconstruction statement: the given fibre functor is an essential hypothesis, not something whose existence we have established.

5. Directed finite-type models without removing nilpotents

We first record a useful identity for this commutative \(H\).

Lemma 5.1. \(S^2=\operatorname{id}_H\).

Proof. Apply \(S\) to each of the two antipode equations. Since \(S\) reverses products and \(H\) is commutative, the resulting equations say \(S^2*S=u\epsilon=S*S^2\). The ordinary identity map also satisfies \(\operatorname{id}_H*S=u\epsilon=S*\operatorname{id}_H\). In an associative convolution algebra two two-sided inverses of the same element are equal: if \(ab=ba=1\) and \(cb=bc=1\), then \(a=a(bc)=(ab)c=c\). Hence \(S^2=\operatorname{id}_H\). \(\square\)

Proposition 5.2. \(H\) is a directed union of finitely generated commutative Hopf subalgebras. Consequently \(G\) is an inverse limit of affine group schemes of finite type, in the precise sense of a pointwise natural limit on all scalar algebras.

Proof. By Finite subcoalgebras, Section 1, every finite set in \(H\) lies in a finite-dimensional subcoalgebra \(C\). Let \(K_C\) be the unital subalgebra generated by \(C+S(C)\). It is finitely generated, since this is finite dimensional. The coproduct of \(C\) lies in \(C\otimes C\), and the anti-coalgebra law puts that of \(S(C)\) in \(S(C)\otimes S(C)\). Multiplicativity therefore puts \(\Delta(K_C)\) in \(K_C\otimes K_C\). Tensoring inclusions of vector spaces is injective, so this really is a restriction of the coproduct. The counit restricts. The anti-algebra law and Lemma 5.1 give \(S(K_C)\subseteq K_C\). All Hopf identities restrict to \(K_C\).

Given two such \(C,D\), their sum is again a finite-dimensional subcoalgebra, and \(K_C,K_D\subseteq K_{C+D}\). Every element of \(H\) lies in one \(C\), so the directed union is all \(H\).

A compatible family of algebra maps \(K_C\to R\) defines a unique map \(H\to R\): define it on an element in any containing \(K_C\), use a common upper bound for independence, and check sums and products in that same upper bound. Restrictions give the inverse construction. The restricted Hopf structures make this a group isomorphism, natural in \(R\):

Finite type means that the coordinate algebra is finitely generated over \(k\). This is exactly what was proved; no claim about reducedness or smoothness is part of it. \(\square\)

Proposition 5.3. Each finite-type model above is a closed affine subgroup functor of some \(\operatorname{GL}_n\).

Proof. Work in a finitely generated commutative Hopf algebra \(K\). Its finite set of algebra generators, together with \(1\), lies in a finite-dimensional subcoalgebra \(D\subseteq K\), by the same finite-subcoalgebra theorem applied to \(K\) itself. In a basis \(d_j\), write \(\Delta(d_j)=\sum_i d_i\otimes t_{ij}\). Contract the first factor by \(\epsilon\) to get \(d_j=\sum_i\epsilon(d_i)t_{ij}\). Thus the coefficients \(t_{ij}\) generate \(K\) as an algebra. The two antipode equations give matrix inverses \(T^{-1}=(S(t_{ij}))\).

Let \[ \begin{gathered} L=k[x_{ij},y_{ij}]\\ /(XY-I,\ YX-I). \end{gathered} \tag{5.2} \] For a scalar algebra \(R\), a map \(L\to R\) is exactly an invertible \(n\)-by-\(n\) matrix \(X\), with its unique inverse \(Y\). Matrix multiplication and inverse give its group functor. The assignments \(x_{ij}\mapsto t_{ij},y_{ij}\mapsto S(t_{ij})\) define a surjective algebra map \(L\to K\). Its induced matrix representation is a group homomorphism by the coproduct identities. The corresponding functor is exactly the subset of matrices satisfying all equations in the kernel of this surjection, over every \(R\): a map out of \(L\) factors through \(K=L/\ker\) precisely when it kills that kernel. This is what closed affine subgroup means. The proof does not replace these equations by their common zeros over fields. \(\square\)

6. Graded exercises with complete solutions

Exercise 1 (basic: gradings and roots of unity)

For finite-dimensional \(\mathbb Z\)-graded vector spaces with the usual tensor grading and forgetful \(F\), compute \(G(R)\). Repeat for \(\mathbb Z/n\)-gradings, including \(n=p=\operatorname{char}(k)>0\).

Solution. A natural \(R\)-linear operator on a graded vector space is scalar \(u_m\in R\) on degree \(m\): the maps from the one-dimensional object in degree \(m\) to every vector of that component force this, and direct sums give the entire operator. Tensor compatibility is \(u_{a+b}=u_a u_b\), and the unit says \(u_0=1\). Taking \(b=-a\) forces \(u_a\) to be a unit. In the integer case \(u_m=u^m\) for a unique \(u\in R^\times\). Thus \(G=\operatorname{Spec}k[t,t^{-1}]\), with group law multiplication of units, \(\Delta t=t\otimes t\), \(\epsilon t=1\), \(S(t)=t^{-1}\). The coefficient of the degree-one object is \(t\), and those of all degrees are its Laurent powers; their distinct grading components also give their linear independence.

To check the coefficient-basis assertion directly, let \(P_m\) be the graded projection onto degree \(m\). The coefficient relation for these projections kills coefficients between different degrees. For \(v\) of degree \(m\), the arrow from the one-dimensional degree-\(m\) object taking \(1\) to \(v\) identifies \([\phi\otimes v]\) with \(\phi(v)\) times that object's coefficient. These coefficients are independent: for each \(m\), the functional \([\phi\otimes v]\mapsto\phi(P_m v)\) respects every graded arrow and takes their values to the distinct coordinate vectors.

For cyclic gradings the same argument has the added relation \(u^n=1\), and every such \(u\) defines the operator \(u_m=u^m\), independent of a representative of \(m\). The coordinate algebra is \(k[t,t^{-1}]/(t^n-1)\); the identical projection argument gives its coefficient basis in residues modulo \(n\). When \(n=p\), put \(z=t-1\). In characteristic \(p\), the binomial coefficients between the endpoints vanish, so \(t^p-1=z^p\). The algebra is \(k[z]/(z^p)\), and \(\Delta z=z\otimes1+1\otimes z+z\otimes z\). Every field-valued point has \(z=0\), but for \(R=k[\varepsilon]/(\varepsilon^2)\), the value \(t=1+\varepsilon\) is a nonidentity point with \(t^p=1\). Thus the single field-valued point does not determine this group functor.

Exercise 2 (intermediate: an invisible additive group)

In characteristic \(p>0\), prove that \(B=k[z]/(z^p)\) with primitive \(z\) is a Hopf algebra. Compute its group functor and a two-dimensional faithful matrix realization. Explain why the abstract group of field-valued points gives the wrong reconstruction.

Solution. On \(k[z]\), set \(\Delta z=z\otimes1+1\otimes z\), \(\epsilon z=0\), \(S(z)=-z\), and extend \(\Delta,\epsilon\) as algebra maps and \(S\) as an algebra map. Coassociativity and the counit identities agree on \(z\), hence on every polynomial. The two antipode composites send \(z\) to zero and \(1\) to \(1\), hence are \(u\epsilon\). In characteristic \(p\), \(\Delta(z^p)=z^p\otimes1+1\otimes z^p\); the ideal \((z^p)\) is preserved by the counit and antipode and descends through both coproduct factors. Thus all identities descend to \(B\).

Maps \(B\to R\) are elements \(r\in R\) with \(r^p=0\); convolution is \(r+s\), identity is zero, and inverse is \(-r\). This differs from the multiplicative coordinate law in Exercise 1, which in the shifted variable has \(r+s+rs\).

On a basis \(e_0,e_1\), put \(\rho(e_0)=e_0\otimes1\), \(\rho(e_1)=e_1\otimes1+e_0\otimes z\). Direct expansion gives the counit and coassociativity, since \(z\) is primitive. Its point action is the matrix with diagonal entries \(1\), upper-right entry \(r\), and lower-left entry \(0\). Addition of the upper-right entries proves the group law, and the entry \(r\) makes this injective for every \(R\). It is the closed subgroup of such unipotent matrices defined by \(r^p=0\).

A field has no nonzero nilpotent, so its point group is trivial, even after passing to an algebraic closure. The displayed two-dimensional comodule is not a trivial comodule: \(e_0\otimes z\ne0\). Taking \(R=k[\varepsilon]/(\varepsilon^2)\) gives the nonzero upper-right entry \(\varepsilon\). Replacing \(G\) by its field-point group would lose this action and this comodule, so would not recover the original representation category with its fibre functor.

Exercise 3 (advanced: constant groups over disconnected scalars)

Let \(\Gamma\) be a finite group and \(H=k^\Gamma\), with \(\Delta f(a,b)=f(ab)\), \(\epsilon f=f(1)\), \(Sf(a)=f(a^{-1})\). Describe \(G(R)\) for arbitrary scalar algebras and its multiplication. Compare \(R=k\) and \(R=k\times k\), without a characteristic restriction.

Solution. Let \(\delta_g\) be the characteristic function of \(g\). The equations \(\delta_g^2=\delta_g\), \(\delta_g\delta_h=0\) for \(g\ne h\), and \(\sum_g\delta_g=1\) present this algebra. Indeed they reduce every polynomial to a unique linear combination of the \(\delta_g\), and evaluating at each group element proves independence. Thus a point is a family \((e_g)\) of orthogonal idempotents of \(R\) summing to \(1\). Conversely such a family defines \(f\mapsto\sum_g f(g)e_g\), a unital algebra map by these same equations.

The coproduct is \(\Delta\delta_g=\sum_{ab=g}\delta_a\otimes\delta_b\). For points \(e,f\), their product is therefore \[ (e*f)_g=\sum_{ab=g}e_a f_b. \tag{6.1} \] Its identity has \(e_1=1\) and all other entries zero; its inverse has entry \(e_{g^{-1}}\). Orthogonality also verifies these identities directly. The Hopf identities hold on each \(\delta_g\) by associativity, identity and inverse in \(\Gamma\), so this computation is valid in every characteristic.

In a nonzero connected algebra, meaning one with only idempotents \(0,1\), such a family has exactly one entry \(1\), so its points are \(\Gamma\). In \(k\times k\), each component independently selects a group element. Hence \(G(k\times k)=\Gamma\times\Gamma\), with componentwise group law. A single fixed choice of an element of \(\Gamma\) describes only the diagonal pairs. The zero algebra has the unique all-zero family, as required. No averaging or semisimplicity enters the argument.

Exercise 4 (synthesis: no single finite faithful fibre)

Let \(H=k[t_1^{\pm1},t_2^{\pm1},\ldots]\), with all \(t_i\) grouplike. Prove that its group functor is a limit of finite-coordinate tori, that \(H\) is not finitely generated as an algebra, and that every finite-dimensional representation factors through a finite-coordinate torus. Deduce that no such representation is faithful on all scalar algebras.

Solution. Each Laurent polynomial uses finitely many variables. The subalgebras \(H_J=k[t_j^{\pm1}:j\in J]\), for finite \(J\), are Hopf subalgebras and have union \(H\). A map \(H\to R\) chooses a unit \(u_i\in R^\times\) independently for every \(i\), since every Laurent polynomial has a unique finite evaluation. Thus \(G(R)=\prod_{i\ge1}R^\times=\varprojlim_J(R^\times)^J\), with coordinatewise multiplication and natural restriction maps.

If finitely many elements generated \(H\), their union of variable supports would be a finite set \(J\). Their generated algebra would lie in \(H_J\). For \(i\notin J\), the Laurent monomial \(t_i\) does not lie there: distinct Laurent monomials form a basis, for multiplying a finite alleged relation by a sufficiently large monomial reduces it to the distinct-monomial independence in an ordinary polynomial ring. This contradicts generation.

For any finite-dimensional comodule, choose a basis. Its finitely many coefficient entries lie in some \(H_J\); the comodule equations then already hold in \(H_J\), since its tensor inclusions into \(H\) are injective. The resulting action factors through the projection \(G\to\operatorname{Spec}(H_J)\). Choose \(i\notin J\) and \(R=k[s,s^{-1}]\). The point with \(u_i=s\) and every other \(u_l=1\) is nonidentity, but its image in that finite-coordinate torus is identity. It therefore acts trivially on the representation. This works even when \(k^\times\) has only one element. Every finite-dimensional representation has a nontrivial kernel as a group functor, though all representations together recover \(G\).

References

[EGNO] Pavel Etingof, Shlomo Gelaki, Dmitri Nikshych and Victor Ostrik, Tensor Categories, Section 5.4, especially Example 5.4.4(i). Freely available author-hosted final version. This is context for reconstruction from affine-group representations. Sections 1–5 above give the arbitrary-scalar natural-operator calculation, representing-group argument, representation/comodule inverse and finite-type construction used here.