Graded (mathematics)

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For other uses of "graded", see Grade.

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 = \oplus_{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 \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.
    • 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-graded vector space or graded linear space is thus a vector space with a decomposition into a direct sum V = \oplus_{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 R_i such that R_i R_j \subseteq R_{i+j}, with i taken from some monoid, usually \mathbb{N} or \mathbb{Z}, or semigroup (for a ring without identity).
  • A graded module is left module M over a graded ring which is a direct sum \oplus_{i \in I} M_i of modules satisfying R_i M_j \subseteq M_{i+j}.
  • 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_iR_j \subseteq A_{i+j} \supseteq  R_iA_j.
  • 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 \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 [,]: L_i \otimes L_j \to L_{i+j} and a differential d: 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:\mathcal{C} \rightarrow \mathcal{A}.
  • Graded manifold – extension of the manifold concept based on ideas coming from supersymmetry and supercommutative algebra, including sections on

In other areas of mathematics: