Английская Википедия:Graded (mathematics)
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Шаблон:Other uses of 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 <math>X</math> is said to be <math>I</math>-graded for an index set <math>I</math> if it has a gradation or grading, i.e. a decomposition into a direct sum <math display="inline">X = \bigoplus_{i \in I} X_i</math> of structures; the elements of <math>X_i</math> are said to be "homogeneous of degree i ".
- The index set <math>I</math> is most commonly <math>\N</math> or <math>\Z</math>, and may be required to have extra structure depending on the type of <math>X</math>.
- Grading by <math>\Z_2</math> (i.e. <math>\Z/2\Z</math>) is also important; see e.g. signed set (the <math>\Z_2</math>-graded sets).
- The trivial (<math>\Z</math>- or <math>\N</math>-) gradation has <math>X_0 = X, X_i = 0</math> for <math>i \neq 0</math> and a suitable trivial structure <math>0</math>.
- 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 <math>I</math>-graded vector space or graded linear space is thus a vector space with a decomposition into a direct sum <math display="inline">V = \bigoplus_{i \in I} V_i</math> 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 <math>R_i</math> such that <math>R_i R_j \subseteq R_{i+j}</math>, with <math>i</math> taken from some monoid, usually <math>\N</math> or <math>\mathbb{Z}</math>, or semigroup (for a ring without identity).
- The associated graded ring of a commutative ring <math>R</math> with respect to a proper ideal <math>I</math> is <math display="inline">\operatorname{gr}_I R = \bigoplus_{n \in \N} I^n/I^{n+1}</math>.
- A graded module is left module <math>M</math> over a graded ring that is a direct sum <math display="inline">\bigoplus_{i \in I} M_i</math> of modules satisfying <math>R_i M_j \subseteq M_{i+j}</math>.
- The associated graded module of an <math>R</math>-module <math>M</math> with respect to a proper ideal <math>I</math> is <math display="inline">\operatorname{gr}_I M = \bigoplus_{n \in \N} I^n M/ I^{n+1} M</math>.
- A differential graded module, differential graded <math>\mathbb{Z}</math>-module or DG-module is a graded module <math>M</math> with a differential <math>d \colon M \to M \colon M_i \to M_{i+1}</math> making <math>M</math> a chain complex, i.e. <math>d \circ d = 0</math> .
- A graded algebra is an algebra <math>A</math> over a ring <math>R</math> that is graded as a ring; if <math>R</math> is graded we also require <math>A_i R_j \subseteq A_{i+j} \supseteq R_iA_j</math>.
- The graded Leibniz rule for a map <math>d\colon A \to A</math> on a graded algebra <math>A</math> specifies that <math>d(a \cdot b) = (da) \cdot b + (-1)^{|a|}a \cdot (db)</math>.
- 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 <math>D(ab) = D(a)b + \varepsilon^{|a||D|}aD(b), \varepsilon = \pm 1</math> acting on homogeneous elements of A.
- A graded derivation is a sum of homogeneous derivations with the same <math>\varepsilon</math>.
- A DGA is an augmented DG-algebra, or differential graded augmented algebra, (see Differential graded algebra).
- A superalgebra is a <math>\mathbb{Z}_2</math>-graded algebra.
- A graded-commutative superalgebra satisfies the "supercommutative" law <math>yx = (-1)^{|x| |y|}xy.</math> for homogeneous x,y, where <math>|a|</math> represents the "parity" of <math>a</math>, 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 <math>\Z_2</math>-gradation.
- A differential graded Lie algebra is a graded vector space over a field of characteristic zero together with a bilinear map <math>[\ , ]\colon L_i \otimes L_j \to L_{i+j}</math> and a differential <math>d\colon L_i \to L_{i-1}</math> satisfying <math>[x,y] = (-1)^{|x||y|+1}[y,x],</math> 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 <math>BW(F)</math> classifying finite-dimensional graded central division algebras over the field F.
- An <math>\mathcal{A}</math>-graded category for a category <math>\mathcal{A}</math> is a category <math>\mathcal{C}</math> together with a functor <math>F\colon \mathcal{C} \rightarrow \mathcal{A}</math>.
- A differential graded category or DG category is a category whose morphism sets form differential graded <math>\mathbb{Z}</math>-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 <math>P</math> with a rank function <math>\rho\colon P \to \N</math> compatible with the ordering (i.e. <math>\rho(x) < \rho(y) \implies x < y</math>) such that <math>y</math> covers <math>x \implies \rho(y) = \rho(x)+1</math> .
