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 $X = \oplus_{i \in I} X_i$ of structures; the elements of $X_i$ are said to be “homogeneous of degree i”.
  • The index set I is most commonly $\mathbb{N}$ or $\mathbb{Z}$, and may be required to have extra structure depending on the type of $X$.
  • Grading by $\mathbb{Z}_2$ (i.e. $\mathbb{Z}/2\mathbb{Z}$) is also important.
  • The trivial ($\mathbb{Z}$- or $\mathbb{N}$-) gradation has $X_0 = X, X_i = 0$ for $i \neq 0$ 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 $V = \oplus_{i \in I} V_i$ 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 $\mathbb{N}$ or $\mathbb{Z}$, or semigroup (for a ring without identity).
  • The associated graded ring of a commutative ring $R$ with respect to a proper ideal $I$ is $\operatorname{gr}_I R = \oplus_{n \in \mathbb{N}} I^n/I^{n+1}$.
• A graded module is left module $M$ over a graded ring which is a direct sum $\oplus_{i \in I} M_i$ 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 $\operatorname{gr}_I M = \oplus_{n \in \mathbb{N}} I^n M/ I^{n+1} M$.
  • A differential graded module, differential graded $\mathbb{Z}$-module or DG-module is a graded module $M$ with a differential $d\colon M \to M \colon M_i \to M_{i+1}$ 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 $A_iR_j \subseteq A_{i+j} \supseteq R_iA_j$.
  • The graded Leibniz rule for a map $d\colon A \to A$ 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 which 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 $D(ab)=D(a)b+\varepsilon^{|a||D|}aD(b), \varepsilon = \pm 1$ acting on homogeneous elements of A.
  • A graded derivation is a sum of homogeneous derivations with the same $\varepsilon$.
  • A DGA is an augmented DG-algebra, or differential graded augmented algebra, (see differential graded algebra).
  • A superalgebra is a $\mathbb{Z}_2$-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 which 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 $\mathbb{Z}_2$-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 $\mathcal{A}$-graded category for a category $\mathcal{A}$ is a category $\mathcal{C}$ together with a functor $F:\mathcal{C} \rightarrow \mathcal{A}$.
  • A differential graded category or DG category is a category whose morphism sets form differential graded $\mathbb{Z}$-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 $\rho\colon P \to \mathbb{N}$ compatible with the ordering (i.e. $\rho(x) < \rho(x) \implies x < y$) such that $y$ covers $x \implies \rho(y)=\rho(x)+1$ .