Graded (mathematics) | Wikipedia audio article


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 {\displaystyle X}
is said to be I {\displaystyle I}
-graded for an index set I {\displaystyle I}
if it has a gradation or grading, i.e. a decomposition into a direct sum X
=⨁ i
∈ I X i {\displaystyle X=\bigoplus _{i\in I}X_{i}}
of structures; the elements of X i {\displaystyle X_{i}}
are said to be “homogeneous of degree i”. The index set I {\displaystyle I}
is most commonly N {\displaystyle \mathbb {N} }
or Z {\displaystyle \mathbb {Z} }
, and may be required to have extra structure depending on the type of X {\displaystyle X}
. Grading by Z 2 {\displaystyle \mathbb {Z} _{2}}
(i.e. Z / 2 Z {\displaystyle \mathbb {Z} /2\mathbb {Z} }
) is also important. The trivial ( Z {\displaystyle \mathbb {Z} }
– or N {\displaystyle \mathbb {N} }
-) gradation has X=
X , X i=
0 {\displaystyle X_{0}=X,X_{i}=0}
for i
≠ {\displaystyle i\neq 0}
and a suitable trivial structure {\displaystyle 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 {\displaystyle I}
-graded vector space or graded linear space is thus a vector space with a decomposition
into a direct sum V
=⨁ i
∈ I V i {\displaystyle 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 abelian groups R i {\displaystyle R_{i}}
such that R i R j ⊆ R i
+ j {\displaystyle R_{i}R_{j}\subseteq R_{i+j}}
, with i {\displaystyle i}
taken from some monoid, usually N {\displaystyle \mathbb {N} }
or Z {\displaystyle \mathbb {Z} }
, or semigroup (for a ring without identity). The associated graded ring of a commutative
ring R {\displaystyle R}
with respect to a proper ideal I {\displaystyle I}
is gr I ⁡
R=⨁ n
∈ N I n / I n
+ 1 {\displaystyle \operatorname {gr} _{I}R=\bigoplus
_{n\in \mathbb {N} }I^{n}/I^{n+1}} .
A graded module is left module M {\displaystyle M}
over a graded ring which is a direct sum ⨁ i
∈ I M i {\displaystyle \bigoplus _{i\in I}M_{i}}
of modules satisfying R i M j ⊆ M i
+ j {\displaystyle R_{i}M_{j}\subseteq M_{i+j}}
. The associated graded module of an R {\displaystyle R}
-module M {\displaystyle M}
with respect to a proper ideal I {\displaystyle I}
is gr I ⁡
M=⨁ n
∈ N I n M / I n
+ 1 M {\displaystyle \operatorname {gr} _{I}M=\bigoplus
_{n\in \mathbb {N} }I^{n}M/I^{n+1}M} .
A differential graded module, differential graded Z {\displaystyle \mathbb {Z} }
-module or DG-module is a graded module M {\displaystyle M}
with a differential d
: M
→ M
: M i → M i
+ 1 {\displaystyle d\colon M\to M\colon M_{i}\to
M_{i+1}} making M {\displaystyle M}
a chain complex, i.e. d
∘ d
={\displaystyle d\circ d=0}
. A graded algebra is an algebra A {\displaystyle A}
over a ring R {\displaystyle R}
that is graded as a ring; if R {\displaystyle R}
is graded we also require A i R j ⊆ A i
+ j ⊇ R i A j {\displaystyle A_{i}R_{j}\subseteq A_{i+j}\supseteq
R_{i}A_{j}} .
The graded Leibniz rule for a map d
: A
→ A {\displaystyle d\colon A\to A}
on a graded algebra A {\displaystyle A}
specifies that d
( a
⋅ b
)=
( d
a )
⋅ b
+ (
− 1 ) | a | a
⋅ (
d b
) {\displaystyle d(a\cdot b)=(da)\cdot b+(-1)^{|a|}a\cdot
(db)} .
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 D
( a
b )
=D
( a
) b
+ ε | a | | D | a
D (
b )
, ε

1 {\displaystyle 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 ε {\displaystyle \varepsilon }
. A DGA is an augmented DG-algebra, or differential
graded augmented algebra, (see differential graded algebra).
A superalgebra is a Z 2 {\displaystyle \mathbb {Z} _{2}}
-graded algebra. A graded-commutative superalgebra satisfies
the “supercommutative” law y
x=
( −
1 ) | x | | y | x
y . {\displaystyle yx=(-1)^{|x||y|}xy.}
for homogeneous x,y, where | a | {\displaystyle |a|}
represents the “parity” of a {\displaystyle 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 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 Z 2 {\displaystyle \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 ⊗ L j → L i
+ j {\displaystyle [,]\colon L_{i}\otimes L_{j}\to
L_{i+j}} and a differential d
: L i → L i
− 1 {\displaystyle d\colon L_{i}\to L_{i-1}}
satisfying [
x ,
y ]
=(
− 1 ) | x | | y | +
1 [
y ,
x ]
, {\displaystyle [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 B
W (
F ) {\displaystyle BW(F)}
classifying finite-dimensional graded central division algebras over the field F.
An A {\displaystyle {\mathcal {A}}}
-graded category for a category A {\displaystyle {\mathcal {A}}}
is a category C {\displaystyle {\mathcal {C}}}
together with a functor F
: C → A {\displaystyle F\colon {\mathcal {C}}\rightarrow
{\mathcal {A}}} .
A differential graded category or DG category is a category whose morphism sets form differential
graded Z {\displaystyle \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 calculusIn other areas
of mathematics: Functionally graded elements are used in finite
element analysis. A graded poset is a poset P {\displaystyle P}
with a rank function ρ
: P
→ N {\displaystyle \rho \colon P\to \mathbb {N}
} compatible with the ordering (i.e. ρ
( x
)

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