Graded (mathematics)

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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 is said to be -graded for an index set if it has a gradation or grading, i.e. a decomposition into a direct sum of structures; the elements of are said to be “homogeneous of degree i”.
    • The index set I is most commonly or , and may be required to have extra structure depending on the type of .
    • Grading by (i.e. ) is also important.
    • The trivial (- or -) gradation has for and a suitable trivial structure .
    • 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 -graded vector space or graded linear space 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 .
    • 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 homogeneous derivation on a graded algebra A is a homogeneous linear map of grade d = |D| on A such that acting on homogeneous elements of A.
    • A graded derivation is a sum of homogeneous derivations with the same .
    • A DGA is an augmented DG-algebra, or differential graded augmented algebra, (see differential graded algebra).
    • A superalgebra is a -graded algebra.
      • A graded-commutative superalgebra satisfies the “supercommutative” law for homogeneous x,y, where represents the “parity” of , 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 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 -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.
  • 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 with a rank function compatible with the ordering (i.e. ) such that covers .