Graded (mathematics)

In mathematics, the term “graded” has a number of meanings, mostly related:

In abstract algebra, it refers to a family of concepts:

* An algebraic structure X is said to be I-graded for an index set I if it has a gradation or grading, i.e. a decomposition into a direct sum X=⨁i∈IXi of structures; the elements of Xi are said to be “homogeneous of degree i”.
* The index set I is most commonly ℕ or ℤ, and may be required to have extra structure depending on the type of X.
* Grading by ℤ2 (i.e. ℤ/2ℤ) is also important; see e.g. signed set (the ℤ2-graded sets).
* The trivial (ℤ\- or ℕ-) gradation has X0=X,Xi=0 for i≠0 and a suitable trivial structure 0.
* An algebraic structure is said to be doubly graded if the index set is a direct product of sets; the pairs may be called “bidegrees” (e.g. see spectral sequence).
* A I-graded vector space or graded linear space is thus a vector space with a decomposition into a direct sum V=⨁i∈IVi of spaces.
* A graded linear map is a map between graded vector spaces respecting their gradations.
* A graded ring is a ring that is a direct sum of abelian groups Ri such that RiRj⊆Ri+j, with i taken from some monoid, usually ℕ or ℤ, or semigroup (for a ring without identity).
* The associated graded ring of a commutative ring R with respect to a proper ideal I is grIR=⨁n∈ℕIn/In+1.
* A graded module is left module M over a graded ring that is a direct sum ⨁i∈IMi of modules satisfying RiMj⊆Mi+j.
* The associated graded module of an R-module M with respect to a proper ideal I is grIM=⨁n∈ℕInM/In+1M.
* A differential graded module, differential graded ℤ-module or DG-module is a graded module M with a differential d:M->M:Mi->Mi+1 making M a chain complex, i.e. d∘d=0 .
* A graded algebra is an algebra A over a ring R that is graded as a ring; if R is graded we also require AiRj⊆Ai+j⊇RiAj.
* The graded Leibniz rule for a map d:A->A on a graded algebra A specifies that d(a⋅b)=(da)⋅b+(−1)|a|a⋅(db) .
* A differential graded algebra, DG-algebra or DGAlgebra is a graded algebra that is a differential graded module whose differential obeys the graded Leibniz rule.
* A homogeneous derivation on a graded algebra A is a homogeneous linear map of grade d = |D| on A such that D(ab)=D(a)b+ε|a||D|aD(b),ε=±1 acting on homogeneous elements of A.
* A graded derivation is a sum of homogeneous derivations with the same ε.
* A DGA is an augmented DG-algebra, or differential graded augmented algebra, (see differential graded algebra).
* A superalgebra is a ℤ2-graded algebra.
* A graded-commutative superalgebra satisfies the “supercommutative” law yx=(−1)|x||y|xy. for homogeneous x,y, where |a| represents the “parity” of a, i.e. 0 or 1 depending on the component in which it lies.
* CDGA may refer to the category of augmented differential graded commutative algebras.
* A graded Lie algebra is a Lie algebra that is graded as a vector space by a gradation compatible with its Lie bracket.
* A graded Lie superalgebra is a graded Lie algebra with the requirement for anticommutativity of its Lie bracket relaxed.
* A supergraded Lie superalgebra is a graded Lie superalgebra with an additional super ℤ2-gradation.
* A differential graded Lie algebra is a graded vector space over a field of characteristic zero together with a bilinear map [,]:Li⊗Lj->Li+j and a differential d:Li->Li−1 satisfying [x,y]=(−1)|x||y|+1[y,x], for any homogeneous elements x, y in L, the “graded Jacobi identity” and the graded Leibniz rule.
* The Graded Brauer group is a synonym for the Brauer–Wall group BW(F) classifying finite-dimensional graded central division algebras over the field F.
* An 𝒜-graded category for a category 𝒜 is a category 𝒞 together with a functor F:𝒞->𝒜.
* A differential graded category or  DG category is a category whose morphism sets form differential graded ℤ-modules.
* Graded manifold – extension of the manifold concept based on ideas coming from supersymmetry and supercommutative algebra, including sections on
* Graded function
* Graded vector fields
* Graded exterior forms
* Graded differential geometry
* Graded differential calculus

In other areas of mathematics:

* Functionally graded elements are used in finite element analysis.
* A graded poset is a poset P with a rank function ρ:P->ℕ compatible with the ordering (i.e. ρ(x)<ρ(y)⟹x<y) such that y covers x⟹ρ(y)=ρ(x)+1 .

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