In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, the term “graded” has a number of meanings, mostly related:
In
abstract algebra
In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures. Algebraic structures include groups, rings, fields, modules, vector spaces, lattices, and algebras over a field. The term ''a ...
, it refers to a family of concepts:
* An
algebraic structure
In mathematics, an algebraic structure consists of a nonempty set ''A'' (called the underlying set, carrier set or domain), a collection of operations on ''A'' (typically binary operations such as addition and multiplication), and a finite set of ...
is said to be
-graded for an
index set
In mathematics, an index set is a set whose members label (or index) members of another set. For instance, if the elements of a set may be ''indexed'' or ''labeled'' by means of the elements of a set , then is an index set. The indexing consists ...
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
is most commonly
or
, and may be required to have extra structure depending on the type of
.
** Grading by
(i.e.
) is also important; see e.g.
signed set (the
-graded sets).
** 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
In mathematics, a graded vector space is a vector space that has the extra structure of a '' grading'' or a ''gradation'', which is a decomposition of the vector space into a direct sum of vector subspaces.
Integer gradation
Let \mathbb be th ...
or graded linear space is thus a
vector space
In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called ''vectors'', may be added together and multiplied ("scaled") by numbers called '' scalars''. Scalars are often real numbers, but can ...
with a decomposition into a
direct sum of spaces.
** A
graded linear map
In mathematics, a graded vector space is a vector space that has the extra structure of a ''graded (mathematics), grading'' or a ''gradation'', which is a decomposition of the vector space into a direct sum of vector spaces, direct sum of linear s ...
is a map between graded vector spaces respecting their gradations.
* A
graded ring
In mathematics, in particular abstract algebra, a graded ring is a ring such that the underlying additive group is a direct sum of abelian groups R_i such that R_i R_j \subseteq R_. The index set is usually the set of nonnegative integers or the ...
is a
ring
Ring may refer to:
* Ring (jewellery), a round band, usually made of metal, worn as ornamental jewelry
* To make a sound with a bell, and the sound made by a bell
:(hence) to initiate a telephone connection
Arts, entertainment and media Film and ...
that is a
direct sum of additive
abelian group
In mathematics, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written. That is, the group operation is comm ...
s
such that
, with
taken from some
monoid
In abstract algebra, a branch of mathematics, a monoid is a set equipped with an associative binary operation and an identity element. For example, the nonnegative integers with addition form a monoid, the identity element being 0.
Monoids ...
, usually
or
, or
semigroup
In mathematics, a semigroup is an algebraic structure consisting of a set together with an associative internal binary operation on it.
The binary operation of a semigroup is most often denoted multiplicatively: ''x''·''y'', or simply ''xy'', ...
(for a
ring without identity).
** The
associated graded ring In mathematics, the associated graded ring of a ring ''R'' with respect to a proper ideal ''I'' is the graded ring:
:\operatorname_I R = \oplus_^\infty I^n/I^.
Similarly, if ''M'' is a left ''R''-module, then the associated graded module is the gra ...
of a
commutative ring with respect to a proper
ideal
Ideal may refer to:
Philosophy
* Ideal (ethics), values that one actively pursues as goals
* Platonic ideal, a philosophical idea of trueness of form, associated with Plato
Mathematics
* Ideal (ring theory), special subsets of a ring considere ...
is
.
* A
graded module
In mathematics, in particular abstract algebra, a graded ring is a ring such that the underlying additive group is a direct sum of abelian groups R_i such that R_i R_j \subseteq R_. The index set is usually the set of nonnegative integers or the ...
is left
module
Module, modular and modularity may refer to the concept of modularity. They may also refer to:
Computing and engineering
* Modular design, the engineering discipline of designing complex devices using separately designed sub-components
* Mo ...
over a graded ring that 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
In mathematics, in particular abstract algebra, a graded ring is a ring such that the underlying additive group is a direct sum of abelian groups R_i such that R_i R_j \subseteq R_. The index set is usually the set of nonnegative integers or the se ...
is an
algebra
Algebra () is one of the broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of almost all of mathematics.
Elementary a ...
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
In mathematics, in particular abstract algebra and topology, a differential graded algebra is a graded associative algebra with an added chain complex structure that respects the algebra structure.
__TOC__
Definition
A differential graded alg ...
, DG-algebra or DGAlgebra is a graded algebra that is a differential graded module whose differential obeys the graded Leibniz rule.
** A
homogeneous derivation In mathematics, a derivation is a function on an algebra which generalizes certain features of the derivative operator. Specifically, given an algebra ''A'' over a ring or a field ''K'', a ''K''-derivation is a ''K''-linear map that satisfies L ...
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
In mathematics, in particular abstract algebra and topology, a differential graded algebra is a graded associative algebra with an added chain complex structure that respects the algebra structure.
__TOC__
Definition
A differential graded a ...
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
In mathematics, in particular abstract algebra and topology, a differential graded algebra is a graded associative algebra with an added chain complex structure that respects the algebra structure.
__TOC__
Definition
A differential graded alg ...
).
** A
superalgebra
In mathematics and theoretical physics, a superalgebra is a Z2-graded algebra. That is, it is an algebra over a commutative ring or field with a decomposition into "even" and "odd" pieces and a multiplication operator that respects the grading.
T ...
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 In mathematics, a graded Lie algebra is a Lie algebra endowed with a gradation which is compatible with the Lie bracket. In other words, a graded Lie algebra is a Lie algebra which is also a nonassociative graded algebra under the bracket oper ...
is a
Lie algebra
In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an Binary operation, operation called the Lie bracket, an Alternating multilinear map, alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow ...
that is graded as a vector space by a gradation compatible with its Lie bracket.
** A
graded Lie superalgebra In mathematics, a graded Lie algebra is a Lie algebra endowed with a gradation which is compatible with the Lie bracket. In other words, a graded Lie algebra is a Lie algebra which is also a nonassociative graded algebra under the bracket oper ...
is a graded Lie algebra with the requirement for anticommutativity of its Lie bracket relaxed.
** A
supergraded Lie superalgebra In mathematics, a graded Lie algebra is a Lie algebra endowed with a graded vector space, gradation which is compatible with the Lie bracket. In other words, a graded Lie algebra is a Lie algebra which is also a nonassociative algebra, nonassociati ...
is a graded Lie superalgebra with an additional super
-gradation.
** A
differential graded Lie algebra
In mathematics, in particular abstract algebra and topology, a differential graded Lie algebra (or dg Lie algebra, or dgla) is a graded vector space with added Lie algebra and chain complex structures that are compatible. Such objects have appl ...
is a graded vector space over a
field
Field may refer to:
Expanses of open ground
* Field (agriculture), an area of land used for agricultural purposes
* Airfield, an aerodrome that lacks the infrastructure of an airport
* Battlefield
* Lawn, an area of mowed grass
* Meadow, a grass ...
of
characteristic zero together with a bilinear map
and a differential
satisfying
for any homogeneous elements ''x'', ''y'' in ''L'', the “graded
Jacobi identity
In mathematics, the Jacobi identity is a property of a binary operation that describes how the order of evaluation, the placement of parentheses in a multiple product, affects the result of the operation. By contrast, for operations with the associ ...
” and the graded Leibniz rule.
* The Graded Brauer group is a synonym for the
Brauer–Wall group In mathematics, the Brauer–Wall group or super Brauer group or graded Brauer group for a field ''F'' is a group BW(''F'') classifying finite-dimensional graded central division algebras over the field. It was first defined by as a generalizatio ...
classifying finite-dimensional graded central
division algebra
In the field of mathematics called abstract algebra, a division algebra is, roughly speaking, an algebra over a field in which division, except by zero, is always possible.
Definitions
Formally, we start with a non-zero algebra ''D'' over a fie ...
s over the field ''F''.
* An
-
graded category
If \mathcal is a category, then a \mathcal-graded category is a category \mathcal together with a functor
F\colon\mathcal \rightarrow \mathcal.
Monoids and groups can be thought of as categories with a single object. A monoid-graded or group-grade ...
for a
category
Category, plural categories, may refer to:
Philosophy and general uses
*Categorization, categories in cognitive science, information science and generally
* Category of being
* ''Categories'' (Aristotle)
* Category (Kant)
* Categories (Peirce) ...
is a category
together with a
functor
In mathematics, specifically category theory, a functor is a Map (mathematics), mapping between Category (mathematics), categories. Functors were first considered in algebraic topology, where algebraic objects (such as the fundamental group) ar ...
.
** A
differential graded category In mathematics, especially homological algebra, a differential graded category, often shortened to dg-category or DG category, is a category whose morphism sets are endowed with the additional structure of a differential graded \Z-module.
In detai ...
or DG category is a category whose morphism sets form differential graded
-modules.
*
Graded manifold
In algebraic geometry, graded manifolds are extensions of the concept of manifold, manifolds based on ideas coming from supersymmetry and supercommutative algebra. Both graded manifolds and supermanifolds are phrased in terms of sheaf (mathematics) ...
– extension of the
manifold
In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a n ...
concept based on ideas coming from supersymmetry and
supercommutative algebra
In mathematics, a supercommutative (associative) algebra is a superalgebra (i.e. a Z2-graded algebra) such that for any two homogeneous elements ''x'', ''y'' we have
:yx = (-1)^xy ,
where , ''x'', denotes the grade of the element and is 0 or 1 ( ...
, including sections on
**
Graded function
**
Graded vector fields
**
Graded exterior forms
**
Graded differential geometry
**
Graded differential calculus
In other areas of mathematics:
*
Functionally graded elements are used in
finite element analysis
The finite element method (FEM) is a popular method for numerically solving differential equations arising in engineering and mathematical modeling. Typical problem areas of interest include the traditional fields of structural analysis, heat ...
.
* A
graded poset
In mathematics, in the branch of combinatorics, a graded poset is a partially-ordered set (poset) ''P'' equipped with a rank function ''ρ'' from ''P'' to the set N of all natural numbers. ''ρ'' must satisfy the following two properties:
* Th ...
is a
poset
In mathematics, especially order theory, a partially ordered set (also poset) formalizes and generalizes the intuitive concept of an ordering, sequencing, or arrangement of the elements of a set. A poset consists of a set together with a binary r ...
with a rank function
compatible with the ordering (i.e.
) such that
covers .
{{sia, mathematics
Linear algebra
Differential operators