calgebras

Calgebras are a class of algebraic structures that extend ordinary algebras by equipping the underlying vector space with a calibration function that assigns to each element a nonnegative integer. This extra datum is intended to organize elements by a grading or filtration and to control how the algebraic operations interact with this structure.

Formally, a calgebra over a field k consists of a k-vector space A equipped with a bilinear product μ: A × A → A and a calibration map κ: A → N0, satisfying compatibility conditions between κ and the product. A common requirement is that κ(xy) ≤ κ(x) + κ(y) for all x, y in A, which encodes a subadditive behavior of the calibration with respect to multiplication. When A has a multiplicative identity, one typically also requests κ(1) = 0. Different variants impose additional constraints, such as equality κ(xy) = κ(x) + κ(y) in graded cases or the use of κ to define a filtration.

Variants of calgebras include graded calgebras, where A decomposes into homogeneous components A = ⊕d Ad with κ(x) equal to the degree of x, and filtered calgebras, where κ induces a filtration by subspaces. Morphisms of calgebras are linear maps that preserve both the product and the calibration.

Examples often cited are polynomial algebras k[x] with κ(f) = deg f, which form graded calgebras, and algebras equipped with the PBW filtration whose associated graded algebra defines a calgebra structure. Calgebras are related to graded and filtered algebras and connect to notions such as associated graded algebras and deformation theory.

See also: graded algebra, filtered algebra, PBW filtration, associated graded algebra.