Graded Automorphism Groups

• Graded automorphism groups are groups of algebra automorphisms that preserve a grading, providing a unified framework for symmetry in algebra and Lie theory.
• They are constructed via scheme-theoretic methods by normalizing diagonal group schemes, and their associated Weyl groups capture permutation symmetries of graded components.
• Explicit computations in settings like associative algebras and quantum affine spaces illustrate their practical importance in analyzing arithmetic phenomena and structural classifications.

A graded automorphism group is the group of algebra automorphisms that respect the structure of a given grading on an algebra or, more generally, the scheme-theoretic group object encoding such automorphisms in the sense of algebraic geometry. This concept serves as a unifying refinement for the paper of symmetries in graded algebraic and Lie-theoretic contexts, underpinning phenomena such as the normalizer-centralizer theory for diagonalizable subgroups, the structure of fine gradings and Weyl group actions, graded-division algebra invariants, and the arithmetic subtleties visible over arbitrary base fields. In its most general formulation, the graded automorphism group is constructed as a group scheme or as a group of invertible elements within suitable bialgebraic duals, accommodating not only algebraically closed fields and classical types but also arbitrary finite-dimensional (possibly nonassociative) algebras over any field.

1. General Definitions and Scheme-Theoretic Structure

Fix a finite-dimensional (not necessarily associative) algebra $A$ over a field $F$, and an abelian group $G$ acting as a grading group. A $G$-grading $\Gamma$ on $A$ is a direct sum decomposition $A = \bigoplus_{g\in G} A_g$ such that $A_g A_h\subset A_{gh}$ for all $g,h\in G$. The graded automorphism group is defined scheme-theoretically via the functor: $$\operatorname{Aut}(\Gamma) : \operatorname{Alg}_F \to \operatorname{Grp}, \quad \operatorname{Aut}(\Gamma)(R) = \{\varphi\in \operatorname{Aut}_{R-\operatorname{alg}}(A\otimes_F R) \mid \text{(permuting or stabilizing the summands)}\}$$ where precise action on the summands is specified by possible permutations, splittings into idempotents, and preservation of the grading (Elduque, 16 Jul 2025).

A diagonalizable affine group scheme $D(G):= \operatorname{Spec} FG$ captures the group-valued characters of $G$, and the grading $\Gamma$ determines a canonical homomorphism $\eta_\Gamma: D(G) \to \operatorname{Aut}(A)$, whose image $\operatorname{Diag}(\Gamma)$ is the "diagonal group scheme" of the grading. The centralizer of $\operatorname{Diag}(\Gamma)$ in $\operatorname{Aut}(A)$ is the stabilizer group scheme of the grading, denoted $\operatorname{Stab}(\Gamma)$. The full automorphism group scheme of the grading coincides with the normalizer of the diagonal group scheme: $$\operatorname{Stab}(\Gamma) = \operatorname{Cent}_{\operatorname{Aut}(A)}(\operatorname{Diag}(\Gamma)), \quad \operatorname{Aut}(\Gamma) = \operatorname{Norm}_{\operatorname{Aut}(A)}(\operatorname{Diag}(\Gamma)).$$ Over algebraically closed fields, this situates all classical automorphism-theoretic results within this general scheme framework (Elduque, 16 Jul 2025).

2. Weyl Groups of Gradings and Quotient Structure

The Weyl group of a grading $\Gamma$ is the group of component-permuting automorphisms modulo componentwise automorphisms: $$W(\Gamma) := \operatorname{Aut}(\Gamma) / \operatorname{Stab}(\Gamma).$$ In the scheme-theoretic context, the Weyl group is the image of a morphism $$\operatorname{Aut}(\Gamma) \to \operatorname{Sym}(\mathrm{supp}(\Gamma))$$ (the finite symmetric group on the support of the grading), and becomes a constant group scheme over $F$ whose geometric points recover the classical Weyl group over an algebraic closure. However, for non-algebraically closed $F$, $W(\Gamma)(F)$ may be smaller than the geometric Weyl group due to arithmetic descent obstructions (Elduque, 16 Jul 2025).

The following exact sequence of group schemes encodes the structure: $$1 \longrightarrow \operatorname{Stab}(\Gamma) \longrightarrow \operatorname{Aut}(\Gamma) \longrightarrow W(\Gamma) \longrightarrow 1.$$ For fine gradings (those with maximal diagonal group), the Weyl group captures all grading symmetries and is functorially invariant under isomorphisms of $A$ as a graded algebra.

3. Explicit Computation and Structure in Key Settings

3.1. Associative Algebras

For associative algebras with abelian $G$-gradings, graded automorphism groups admit explicit block-matrix or tensor factorizations:

• If $A \cong M_k(F) \otimes D$, where $D$ is a graded-division algebra, then

$$\operatorname{Aut}_G(A) \simeq (F^\times)^{k-1} \times \operatorname{Aut}_G(D)$$

is the graded inner-automorphism group (from diagonal scalings and graded-division automorphisms), and the full graded automorphism group incorporates a semidirect product with permutations and outer automorphisms (Rodrigo-Escudero, 2020, Elduque et al., 2010).

3.2. Quantum Affine Space

For quantum affine $n$-space with multiparameter relations, the graded automorphism group is

$$\operatorname{Aut}_{\mathrm{gr}}(A_q) \cong D \rtimes S$$

where $D$ is the product of general linear groups on blocks determined by equivalence of grading parameters (identical rows of the quantum matrix), and $S$ is a permutation group acting by block reordering (Jin, 14 Feb 2025, Jensen et al., 21 Nov 2025). This structure is exhaustive up through dimension 7 and reflects the combinatorial data of block partitions and quantum parameters (Jensen et al., 21 Nov 2025).

3.3. Polynomial Algebras with Grading

For $A = k[x, y, z]$ with a nontrivial $\mathbb{Z}$-grading, the graded automorphism group is generated by diagonal torus automorphisms, certain explicit "upper triangular" automorphisms, and wild automorphisms subject to arithmetic constraints on the grading weights. The group structure is expressed as explicit semidirect products and is determined combinatorially by the degrees of the variables (Trushin, 2022).

3.4. Upper-Triangular Matrix Algebras

For $\mathrm{UT}_n(F)$ under an elementary $G$-grading, all graded automorphisms are inner (by multiplication by a homogeneous invertible element) or, in the Jordan/Lie cases, explicitly decompose into involutive and diagonal factors; the Weyl group structure is trivial in the associative case and at most cyclic of order 2 otherwise (Yasumura, 2017).

3.5. Cross Product Structures and Exceptional Algebras

For multilinear cross product algebras (e.g., octonions, Albert algebra), graded automorphism groups are constructed as affine group schemes with only a handful of possible grading types. The full automorphism scheme aligns precisely with the structure group dictated by the cross product — e.g., type $G_2$ for imaginary octonions, $\mathrm{Spin}_7$ for the third cross product. Classifications of fine gradings and Weyl group computations are given in these cases and are directly linked to root system automorphisms and classical extended Weyl groups (Daza-García et al., 2020, Elduque et al., 2010).

4. Field Dependence and Arithmetic Phenomena

Unlike over algebraically closed fields, where the geometric points of scheme-theoretic graded automorphism and Weyl group coincide with their classical counterparts, over arbitrary base fields the correspondence is partial. For instance, a grading over $A = \mathbb{Q}(\sqrt[3]{2})$ may have a trivial automorphism group and Weyl group over $\mathbb{Q}$, but over $\bar{\mathbb{Q}}$ the Weyl group becomes nontrivial ($C_2$) due to splitting in the base change (Elduque, 16 Jul 2025). In general, $W(\Gamma)$ is a constant group scheme whose set of $F$-rational points can be strictly smaller than the geometric fiber.

A plausible implication is that arithmetic of the base field imposes additional restrictions on permissible permutations and symmetries of the grading, affecting the realization of Weyl group elements and possible graded-isomorphism types.

5. Applications and Broader Connections

5.1. Free Groups, Lie Algebras, and Filtration Structures

In automorphism groups of free groups, the graded automorphism Lie algebra encodes the Andreadakis filtration, and the associated graded Lie algebra structure is controlled by the action of the graded automorphism group (Bartholdi, 2013). Similarly, graded filtrations of representation algebras of free or abelian groups use graded automorphism group actions in Johnson homomorphism theory, allowing extension and generalization of classical cohomological invariants (Satoh, 2016).

5.2. Cluster Algebras and Pseudo-Grading Structures

In cluster algebra theory, the cluster automorphism group can admit a pseudo-$\mathbb{N}$-grading structure, and the generation of the automorphism group by low-weight elements is an instance of a filtered group action compatible with the grading formalism (Fu et al., 28 Mar 2025).

5.3. Crossed Structures and Double Automorphisms

For graded Lie algebras, double automorphism groups arise as automorphisms that permute grading components according to a group automorphism of $A$, forming an extension of the grading-preserving automorphism group by the group of grading automorphisms. This semidirect product structure is fundamental in applications such as selective nilpotency criteria, Frobenius automorphism groups, and the analysis of symmetry for graded Lie rings of groups (1207.1323).

6. Fine Gradings and Invariance Properties

Fine gradings (maximal refinements) have automorphism-theoretic characterizations in terms of maximal diagonalizable group schemes and Weyl group invariance. For graded-simple associative algebras with abelian grading and DCC on graded left ideals, the automorphism group of a fine grading is given explicitly via graded automorphisms of the division component and block-permutation subgroups. The Weyl group is then a direct product of permutation and group-theoretic factors dependent on the refined graded-division algebra structure (Rodrigo-Escudero, 2020, Elduque et al., 2010).

In exceptional algebraic contexts (octonions, Albert algebra), fine gradings and their Weyl groups correspond to root system automorphism groups or classical finite groups ($GL_n(q)$, $SL_n(q)$, affine Weyl groups), demonstrating the centrality of graded automorphism groups in higher algebraic and Lie-theoretic symmetry (Elduque et al., 2010).

7. Classification Frameworks and Categorical Perspectives

In the most general setting, the classification of gradings and their automorphism groups on finite-dimensional algebras reduces to computation in the finite dual bialgebra of the quantum symmetry semigroup $a(A)$, as per the Manin–Tambara construction. Graded automorphisms are the invertible group-like elements in this dual, and isomorphism classes of gradings correspond to bialgebra maps up to conjugation by these invertibles (Militaru, 2021). This framework yields a unifying categorical perspective for both classical and quantum grading symmetries and connects with universal group schemes in the scheme-theoretic realization.

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Summary Table: Scheme-Theoretic Graded Automorphism Constructions

Context	Group Structure	Key Quotient/Weyl Group
General graded algebra $A$	$\operatorname{Aut}(\Gamma)$ scheme	$$W(\Gamma) = \operatorname{Aut}(\Gamma) / \operatorname{Stab}(\Gamma)$$ (Elduque, 16 Jul 2025)
Matrix algebras	$(F^\times)^{k-1} \times \operatorname{Aut}_G(D)$, permutations	$$(T^{k-1}/\Delta(T)) \rtimes (S_k \times \operatorname{Aut}(T,\beta))$$ (Rodrigo-Escudero, 2020, Elduque et al., 2010)
Quantum affine spaces	$D \rtimes S$, blocks and permutations	Orbit-stabilizer determined by quantum parameters (Jin, 14 Feb 2025, Jensen et al., 21 Nov 2025)
Cross product algebras	Classical group schemes ($G_2$, $\mathrm{Spin}_7$, etc.)	Weyl group as root system automorphism group (Daza-García et al., 2020, Elduque et al., 2010)

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References:

• "Automorphism group schemes and Weyl groups of gradings" (Elduque, 16 Jul 2025)
• "Graded automorphisms of quantum affine spaces" (Jin, 14 Feb 2025)
• "Stabilizing Automorphisms of Quantum Affine Space" (Jensen et al., 21 Nov 2025)
• "On the graded automorphisms of upper triangular matrix algebras" (Yasumura, 2017)
• "Fine gradings and automorphism groups on associative algebras" (Rodrigo-Escudero, 2020)
• "Weyl groups of fine gradings on matrix algebras, octonions and the Albert algebra" (Elduque et al., 2010)
• "Cross products, automorphisms, and gradings" (Daza-García et al., 2020)
• "The automorphisms group and the classification of gradings of finite dimensional associative algebras" (Militaru, 2021)
• "Double automorphisms of graded Lie algebras" (1207.1323)
• "Automorphisms of free groups, I" (Bartholdi, 2013)
• "On the graded quotients of the SL(m,ℂ)-representation algebras of groups" (Satoh, 2016).