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 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 = \bigoplus_{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 \N or \Z, and may be required to have extra structure depending on the type of X.
  • Grading by \Z_2 (i.e. \Z/2\Z) is also important; see e.g. signed set (the \Z_2-graded sets).
  • The trivial (\Z- or \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 = \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 such that R_i R_j \subseteq R_{i+j}, with i taken from some monoid, usually \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 = \bigoplus_{n \in \N} I^n/I^{n+1}.
• A graded module is left module M over a graded ring that is a direct sum \bigoplus_{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 = \bigoplus_{n \in \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_i R_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 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-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 \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}-graded category for a category \mathcal{A} is a category \mathcal{C} together with a functor 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
  • 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 \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 .