Associated Graded Construction

A functorial procedure that converts a filtered algebraic object (e.g., filtered algebra, module, Lie algebra) into a graded object by taking successive quotients gr_F(V) = ⊕_i F_i V / F_{i-1} V. The associated graded records leading-order behaviour of the filtration and often simplifies multiplicative relations by passing to graded components.

Operation that converts a filtered object into a graded object by taking successive quotients Gr(F) = ⊕_i F_i / F_{i-1}, thereby encoding the filtration as a direct‑sum of layers.