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} }
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}}
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} .
-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 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

=\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
A superalgebra is a Z 2 {\displaystyle \mathbb {Z} _{2}}
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}}
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