Graded structure

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 $X$ $X$ is said to be $I$ $I$ -graded for an index set $I$ $I$ if it has a gradation or grading, i.e. a decomposition into a direct sum ${\textstyle X=\bigoplus _{i\in I}X_{i}}$ ${\textstyle X=\bigoplus _{i\in I}X_{i}}$ of structures; the elements of $X_{i}$ $X_{i}$ 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; see e.g. signed set (the $\mathbb {Z} _{2}$ -graded sets).
- 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 $I$ $I$ -graded vector space or graded linear space is thus a vector space with a decomposition into a direct sum ${\textstyle V=\bigoplus _{i\in I}V_{i}}$ ${\textstyle V=\bigoplus _{i\in I}V_{i}}$ 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 additive abelian groups $R_{i}$ $R_{i}$ such that $R_{i}R_{j}\subseteq R_{i+j}$ $R_{i}R_{j}\subseteq R_{i+j}$ , with $i$ $i$ taken from some monoid, usually $\mathbb {N}$ $\mathbb {N}$ or $\mathbb {Z}$ $\mathbb {Z}$ , or semigroup (for a ring without identity).
- The associated graded ring of a commutative ring $R$ with respect to a proper ideal $I$ is ${\textstyle \operatorname {gr} _{I}R=\bigoplus _{n\in \mathbb {N} }I^{n}/I^{n+1}}$ .
A graded module is left module $M$ $M$ over a graded ring that is a direct sum ${\textstyle \bigoplus _{i\in I}M_{i}}$ ${\textstyle \bigoplus _{i\in I}M_{i}}$ of modules satisfying $R_{i}M_{j}\subseteq M_{i+j}$ $R_{i}M_{j}\subseteq M_{i+j}$ .
- The associated graded module of an $R$ -module $M$ with respect to a proper ideal $I$ is ${\textstyle \operatorname {gr} _{I}M=\bigoplus _{n\in \mathbb {N} }I^{n}M/I^{n+1}M}$ .
- 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 $A$ $A$ over a ring $R$ $R$ that is graded as a ring; if $R$ $R$ is graded we also require $A_{i}R_{j}\subseteq A_{i+j}\supseteq R_{i}A_{j}$ $A_{i}R_{j}\subseteq A_{i+j}\supseteq R_{i}A_{j}$ .
- 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 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 $D(ab)=D(a)b+\varepsilon ^{|a||D|}aD(b),\varepsilon =\pm 1$ 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 $\mathbb {Z} _{2}$ $\mathbb {Z} _{2}$ -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 $\mathbb {Z} _{2}$ -gradation.
- A differential graded Lie algebra is a graded vector space over a field of characteristic zero together with a bilinear map $[\ ,]\colon L_{i}\otimes L_{j}\to L_{i+j}$ and a differential $d\colon L_{i}\to L_{i-1}$ 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 ${\mathcal {A}}$ ${\mathcal {A}}$ -graded category for a category ${\mathcal {A}}$ ${\mathcal {A}}$ is a category ${\mathcal {C}}$ ${\mathcal {C}}$ together with a functor $F\colon {\mathcal {C}}\rightarrow {\mathcal {A}}$ $F\colon {\mathcal {C}}\rightarrow {\mathcal {A}}$ .
- 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

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. $\rho (x)<\rho (y)\implies x<y$ ) such that $y$ covers $x\implies \rho (y)=\rho (x)+1$ .

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