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 $X=\underset{i\in I}{⨁}{X}_i$ of structures; the elements of $X_i$ are said to be “homogeneous of degree i”.
  • The index set $I$ is most commonly $\mathbb{\mathbb{N}}$ or $\mathbb{\mathbb{Z}}$, and may be required to have extra structure depending on the type of $X$.
  • Grading by ${\mathbb{\mathbb{Z}}}_2$ (i.e. $\mathbb{\mathbb{Z}}/2\mathbb{\mathbb{Z}}$) is also important; see e.g. signed set (the ${\mathbb{\mathbb{Z}}}_2$-graded sets).
  • The trivial ($\mathbb{\mathbb{Z}}$- or $\mathbb{\mathbb{N}}$-) gradation has $X_0=X,X_i=0$ for $i\ne 0$ 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 $V=\underset{i\in I}{⨁}{V}_i$ 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 $R_i$ such that $R_i{R}_j\subseteq R_{i+j}$, with $i$ taken from some monoid, usually $\mathbb{\mathbb{N}}$ or $\mathbb{\mathbb{Z}}$, or semigroup (for a ring without identity).
  • The associated graded ring of a commutative ring $R$ with respect to a proper ideal $I$ is ${\mathrm{gr}}_{I}R=\underset{n\in \mathbb{\mathbb{N}}}{⨁}{I}^n/I^{n+1}$.
• A graded module is left module $M$ over a graded ring that is a direct sum $\underset{i\in I}{⨁}{M}_i$ of modules satisfying $R_i{M}_j\subseteq M_{i+j}$.
  • The associated graded module of an $R$-module $M$ with respect to a proper ideal $I$ is ${\mathrm{gr}}_{I}M=\underset{n\in \mathbb{\mathbb{N}}}{⨁}{I}^{n}M/I^{n+1}M$.
  • A differential graded module, differential graded $\mathbb{\mathbb{Z}}$-module or DG-module is a graded module $M$ with a differential $d:M\to M:M_i\to M_{i+1}$ making $M$ a chain complex, i.e. $d\circ d=0$ .
• A graded algebra is an algebra $A$ over a ring $R$ that is graded as a ring; if $R$ is graded we also require $A_i{R}_j\subseteq A_{i+j}\supseteq R_i{A}_j$.
  • The graded Leibniz rule for a map $d:A\to A$ on a graded algebra $A$ specifies that $d(a\cdot b)=(da)\cdot b+(-1{)}^{|a|}a\cdot (db)$ .
  • 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 $D(ab)=D(a)b+{\epsilon }^{|a||D|}aD(b),\epsilon =\pm 1$ acting on homogeneous elements of A.
  • A graded derivation is a sum of homogeneous derivations with the same $\epsilon$.
  • A DGA is an augmented DG-algebra, or differential graded augmented algebra, (see differential graded algebra).
  • A superalgebra is a ${\mathbb{\mathbb{Z}}}_2$-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 ${\mathbb{\mathbb{Z}}}_2$-gradation.
  • A differential graded Lie algebra is a graded vector space over a field of characteristic zero together with a bilinear map $[,]:L_i\otimes L_j\to L_{i+j}$ and a differential $d:L_i\to L_{i-1}$ 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 $F:{𝒞}\to {𝒜}$.
  • A differential graded category or DG category is a category whose morphism sets form differential graded $\mathbb{\mathbb{Z}}$-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 $\rho :P\to \mathbb{\mathbb{N}}$ compatible with the ordering (i.e. $\rho (x)<\rho (y)⟹x<y$) such that $y$ covers $x⟹\rho (y)=\rho (x)+1$ .