# Graded (mathematics)

Jump to navigation
Jump to search

Template:Disambiguation
In mathematics, the term “**graded**” has a number of related meanings:

- An algebraic structure 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 of structures; the elements of are said to be “homogenous of degree*i*”.- The index set I is most commonly or , and may be required to have extra structure depending on the type of .
- The
**trivial**(- or -) gradation has , for**Failed to parse (syntax error): {\displaystyle i ≠ 0}**and a suitable trivial structure . - An algebraic structure is said to be doubly graded if the index set is a direct product; the pairs may be called as “bidegrees” (e.g. see spectral sequence).

- A I-graded vector space or
**graded linear space**for a set I is thus a vector space with a decomposition into a direct sum 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 such that , with taken from some monoid, usually or , or semigroup (for a ring without identity).
- The associated graded ring of a commutative ring with respect to a proper ideal is .

- A graded module is left module over a graded ring which is a direct sum of modules satisfying .
- The associated graded module of an -module with respect to a proper ideal is .
- A differential graded module,
**differential graded -module**or**DG-module**is a graded module with a**differential**making a**chain complex**, i.e. .

- A graded algebra is an algebra over a ring that is graded as a ring; if is graded we also require
**Failed to parse (syntax error): {\displaystyle A_iR_j \subseteq A_{i+j} ⊇ R_iA_j}**.- The graded Leibniz rule for a map on a graded algebra specifies that .
- 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
**DGA**is an augmented DG-algebra, or**differential graded augmented algebra**, (see differential graded algebra). - A superalgebra is a a
**Z**_{2}-graded algebra.- A graded-commutative superalgebra satisfies the “supercommutative” law for homogenous
*x*,*y*, where represents the “parity” of , i.e. 0 or 1 depending on the component it lies in.

- A graded-commutative superalgebra satisfies the “supercommutative” law for homogenous
**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
**Z**/2**Z**-gradation. - A Differential graded Lie algebra is a graded vector space over a field of characteristic zero together with a bilinear map and a differential satisfying 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 classifying finite-dimensional graded central division algebras over the field*F*. - An -graded category for a category is a category together with a functor .
- A differential graded category or
**DG category**is a category whose morphism sets form differential graded -modules.

- A differential graded category or
- Graded manifold – extension of the manifold concept based on ideas coming from supersymmetry and supercommutative algebra, including sections on
- Graded derivation ,
*but see also the section Graded derivations in Derivation (abstract algebra)* - Functionally graded elements are elements used in finite element analysis.
- A graded poset is a poset
*P*with a**rank function****Failed to parse (syntax error): {\displaystyle ρ\colon P \to N}**compatible with the ordering (so ρ(*x*)<ρ(*y*) ⇐*x*<*y*) such that*y*covers*x*⇒ .