dautomorphismes

Dautomorphismes, or d-automorphisms, are a family of automorphisms constrained by a degree parameter d in various mathematical contexts. The term is not universally standardized and its exact meaning depends on the object and the chosen notion of degree, grading, or filtration.

In graded structures such as a graded vector space V = ⊕_{n∈Z} V_n, a d-automorphism is typically an automorphism f: V → V that preserves the grading up to a bound d, meaning f(V_n) ⊆ ⊕_{m≤n+d} V_m and, conversely, f^{-1}(V_n) ⊆ ⊕_{m≤n+d} V_m. This captures the idea that the action respects degree while allowing controlled deviation controlled by d.

In filtered objects with a filtration F_0 ⊆ F_1 ⊆ ... ⊆ X, a d-automorphism may be required to map F_i into F_{i+d} for all i. This expresses a similar notion of compatibility with a hierarchical structure, where the degree parameter bounds how far the image can shift along the filtration.

In other settings, d-automorphisms may refer to automorphisms of systems defined by degree-d polynomials or to symmetries that preserve a d-skeleton in a combinatorial structure. The exact interpretation is context-dependent, and the terminology is more of a descriptive aid than a fixed standard.

Properties depend on the formal definition; sometimes the set of d-automorphisms forms a subgroup of the full automorphism group, while in other formulations the bound is not preserved by composition. See also: automorphism, graded algebra, filtered algebra, degree, skeleton.