Algebrad

Algebrad is a term used in some mathematical discussions to describe a differential graded algebra–like structure that combines a graded algebra with a compatible differential operator.

Formally, an algebrad over a field K is a tuple (A, ⊕, ·, deg, δ) where A = ⨁_{i∈Z} A_i is a Z-graded K-algebra with multiplication that respects grading (deg(a b) = deg a + deg b for homogeneous a and b). The operator δ: A → A is K-linear and has degree +1, meaning δ(A_i) ⊆ A_{i+1}, with δ^2 = 0. It also satisfies the graded Leibniz rule for homogeneous a: δ(a b) = δ(a) b + (-1)^{deg a} a δ(b). The cohomology H(A, δ) then inherits a graded K-algebra structure.

Properties and relations: If δ is a differential with the graded Leibniz rule, A behaves like a differential graded algebra (DGA). The grading encodes a notion of degree, while δ encodes a differential, allowing formation of cohomology groups that reflect both algebraic and differential data.

Examples: The de Rham algebra Ω^*(M) of differential forms on a smooth manifold M with the exterior derivative d is a canonical example, where deg is the form degree and d^2 = 0. Another basic example is a polynomial algebra K[x] with deg(x) = 1 and δ = 0, giving a trivial differential that still fits the structure.

History and usage: The term algebrad is not widely standardized and largely overlaps with the well-established concept of a differential graded algebra (DGA). In some contexts, authors use algebrad to emphasize the combination of grading and differential features. See also differential graded algebra, graded algebra, and homological algebra.