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