Graded structure explained

In mathematics, the term "graded" has a number of meanings, mostly related:

In abstract algebra, it refers to a family of concepts:

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 $X\; =\; \backslash bigoplus\_\; X\_i$ 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 ≠ 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 $V\; =\; \backslash bigoplus\_\; V\_i$ 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 Z , or semigroup (for a ring without identity).R with respect to a proper ideal I is $\backslash operatorname\_I\; R\; =\; \backslash bigoplus\_\; I^n/I^$.M over a graded ring that is a direct sum $\backslash bigoplus\_\; M\_i$ 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 $\backslash operatorname\_I\; M\; =\; \backslash bigoplus\_\; I^n\; M/\; I^\; M$.

• • A differential graded module, differential graded

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 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 ⋅ b) = (da) ⋅ b + ( -1 ) ^{|a|} a ⋅ (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

Z _{2} -graded algebra.

• • • A graded-commutative superalgebra satisfies the "supercommutative" law

yx = ( -1 ) ^{|x|} 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} ⊗ 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.BW(F) classifying finite-dimensional graded central division algebras over the field F.

• An

l{A} -graded category for a category l{A} is a category l{C} together with a functor F\colon l{C} → l{A} .

• • A differential graded category or DG category is a category whose morphism sets form differential graded

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.

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 .