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 [math]\displaystyle{ X }[/math] is said to be [math]\displaystyle{ I }[/math]-graded for an index set [math]\displaystyle{ I }[/math] if it has a gradation or grading, i.e. a decomposition into a direct sum [math]\displaystyle{ X = \bigoplus_{i \in I} X_i }[/math] of structures; the elements of [math]\displaystyle{ X_i }[/math] are said to be “homogeneous of degree i”.
  • The index set [math]\displaystyle{ I }[/math] is most commonly [math]\displaystyle{ \N }[/math] or [math]\displaystyle{ \Z }[/math], and may be required to have extra structure depending on the type of [math]\displaystyle{ X }[/math].
  • Grading by [math]\displaystyle{ \Z_2 }[/math] (i.e. [math]\displaystyle{ \Z/2\Z }[/math]) is also important; see e.g. signed set (the [math]\displaystyle{ \Z_2 }[/math]-graded sets).
  • The trivial ([math]\displaystyle{ \Z }[/math]- or [math]\displaystyle{ \N }[/math]-) gradation has [math]\displaystyle{ X_0 = X, X_i = 0 }[/math] for [math]\displaystyle{ i \neq 0 }[/math] and a suitable trivial structure [math]\displaystyle{ 0 }[/math].
  • 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 [math]\displaystyle{ I }[/math]-graded vector space or graded linear space is thus a vector space with a decomposition into a direct sum [math]\displaystyle{ V = \bigoplus_{i \in I} V_i }[/math] 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 [math]\displaystyle{ R_i }[/math] such that [math]\displaystyle{ R_i R_j \subseteq R_{i+j} }[/math], with [math]\displaystyle{ i }[/math] taken from some monoid, usually [math]\displaystyle{ \N }[/math] or [math]\displaystyle{ \mathbb{Z} }[/math], or semigroup (for a ring without identity).
  • The associated graded ring of a commutative ring [math]\displaystyle{ R }[/math] with respect to a proper ideal [math]\displaystyle{ I }[/math] is [math]\displaystyle{ \operatorname{gr}_I R = \bigoplus_{n \in \N} I^n/I^{n+1} }[/math].
• A graded module is left module [math]\displaystyle{ M }[/math] over a graded ring that is a direct sum [math]\displaystyle{ \bigoplus_{i \in I} M_i }[/math] of modules satisfying [math]\displaystyle{ R_i M_j \subseteq M_{i+j} }[/math].
  • The associated graded module of an [math]\displaystyle{ R }[/math]-module [math]\displaystyle{ M }[/math] with respect to a proper ideal [math]\displaystyle{ I }[/math] is [math]\displaystyle{ \operatorname{gr}_I M = \bigoplus_{n \in \N} I^n M/ I^{n+1} M }[/math].
  • A differential graded module, differential graded [math]\displaystyle{ \mathbb{Z} }[/math]-module or DG-module is a graded module [math]\displaystyle{ M }[/math] with a differential [math]\displaystyle{ d\colon M \to M \colon M_i \to M_{i+1} }[/math] making [math]\displaystyle{ M }[/math] a chain complex, i.e. [math]\displaystyle{ d \circ d=0 }[/math] .
• A graded algebra is an algebra [math]\displaystyle{ A }[/math] over a ring [math]\displaystyle{ R }[/math] that is graded as a ring; if [math]\displaystyle{ R }[/math] is graded we also require [math]\displaystyle{ A_iR_j \subseteq A_{i+j} \supseteq R_iA_j }[/math].
  • The graded Leibniz rule for a map [math]\displaystyle{ d\colon A \to A }[/math] on a graded algebra [math]\displaystyle{ A }[/math] specifies that [math]\displaystyle{ d(a \cdot b) = (da) \cdot b + (-1)^{|a|}a \cdot (db) }[/math] .
  • 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 [math]\displaystyle{ D(ab)=D(a)b+\varepsilon^{|a||D|}aD(b), \varepsilon = \pm 1 }[/math] acting on homogeneous elements of A.
  • A graded derivation is a sum of homogeneous derivations with the same [math]\displaystyle{ \varepsilon }[/math].
  • A DGA is an augmented DG-algebra, or differential graded augmented algebra, (see differential graded algebra).
  • A superalgebra is a [math]\displaystyle{ \mathbb{Z}_2 }[/math]-graded algebra.
    • A graded-commutative superalgebra satisfies the “supercommutative” law [math]\displaystyle{ yx = (-1)^{|x| |y|}xy. }[/math] for homogeneous x,y, where [math]\displaystyle{ |a| }[/math] represents the “parity” of [math]\displaystyle{ a }[/math], 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 [math]\displaystyle{ \Z_2 }[/math]-gradation.
  • A differential graded Lie algebra is a graded vector space over a field of characteristic zero together with a bilinear map [math]\displaystyle{ [,]\colon L_i \otimes L_j \to L_{i+j} }[/math] and a differential [math]\displaystyle{ d\colon L_i \to L_{i-1} }[/math] satisfying [math]\displaystyle{ [x,y] = (-1)^{|x||y|+1}[y,x], }[/math] 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 [math]\displaystyle{ BW(F) }[/math] classifying finite-dimensional graded central division algebras over the field F.
• An [math]\displaystyle{ \mathcal{A} }[/math]-graded category for a category [math]\displaystyle{ \mathcal{A} }[/math] is a category [math]\displaystyle{ \mathcal{C} }[/math] together with a functor [math]\displaystyle{ F\colon \mathcal{C} \rightarrow \mathcal{A} }[/math].
  • A differential graded category or DG category is a category whose morphism sets form differential graded [math]\displaystyle{ \mathbb{Z} }[/math]-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 [math]\displaystyle{ P }[/math] with a rank function [math]\displaystyle{ \rho\colon P \to \N }[/math] compatible with the ordering (i.e. [math]\displaystyle{ \rho(x) \lt \rho(y) \implies x \lt y }[/math]) such that [math]\displaystyle{ y }[/math] covers [math]\displaystyle{ x \implies \rho(y)=\rho(x)+1 }[/math] .