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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 ...
, a graded vector space is 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 ...
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 the set of non-negative
integer An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign (−1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language ...
s. An \mathbb-graded vector space, often called simply a graded vector space without the prefix \mathbb, is a vector space together with a decomposition into a direct sum of the form : V = \bigoplus_ V_n where each V_n is a vector space. For a given ''n'' the elements of V_n are then called homogeneous elements of degree ''n''. Graded vector spaces are common. For example the set of all
polynomial In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An exa ...
s in one or several variables forms a graded vector space, where the homogeneous elements of degree ''n'' are exactly the linear combinations of
monomial In mathematics, a monomial is, roughly speaking, a polynomial which has only one term. Two definitions of a monomial may be encountered: # A monomial, also called power product, is a product of powers of variables with nonnegative integer expone ...
s of
degree Degree may refer to: As a unit of measurement * Degree (angle), a unit of angle measurement ** Degree of geographical latitude ** Degree of geographical longitude * Degree symbol (°), a notation used in science, engineering, and mathematics ...
 ''n''.


General gradation

The subspaces of a graded vector space need not be indexed by the set of natural numbers, and may be indexed by the elements of any set ''I''. An ''I''-graded vector space ''V'' is a vector space together with a decomposition into a direct sum of subspaces indexed by elements ''i'' of the set ''I'': : V = \bigoplus_ V_i. Therefore, an \mathbb-graded vector space, as defined above, is just an ''I''-graded vector space where the set ''I'' is \mathbb (the set of
natural number In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country"). Numbers used for counting are called ''Cardinal n ...
s). The case where ''I'' is the
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 ...
\mathbb/2\mathbb (the elements 0 and 1) is particularly important in
physics Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which r ...
. A (\mathbb/2\mathbb)-graded vector space is also known as a supervector space.


Homomorphisms

For general index sets ''I'', a
linear map In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a mapping V \to W between two vector spaces that pr ...
between two ''I''-graded vector spaces is called a graded linear map if it preserves the grading of homogeneous elements. A graded linear map is also called a homomorphism (or morphism) of graded vector spaces, or homogeneous linear map: :f(V_i)\subseteq W_i for all ''i'' in ''I''. For a fixed
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 ...
and a fixed index set, the graded vector spaces form 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) ...
whose morphisms are the graded linear maps. When ''I'' is a
commutative In mathematics, a binary operation is commutative if changing the order of the operands does not change the result. It is a fundamental property of many binary operations, and many mathematical proofs depend on it. Most familiar as the name o ...
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 ...
(such as the natural numbers), then one may more generally define linear maps that are homogeneous of any degree ''i'' in ''I'' by the property :f(V_j)\subseteq W_ for all ''j'' in ''I'', where "+" denotes the monoid operation. If moreover ''I'' satisfies the
cancellation property In mathematics, the notion of cancellative is a generalization of the notion of invertible. An element ''a'' in a magma has the left cancellation property (or is left-cancellative) if for all ''b'' and ''c'' in ''M'', always implies that . An ...
so that it can be embedded into an
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 ...
''A'' that it generates (for instance the integers if ''I'' is the natural numbers), then one may also define linear maps that are homogeneous of degree ''i'' in ''A'' by the same property (but now "+" denotes the group operation in ''A''). Specifically, for ''i'' in ''I'' a linear map will be homogeneous of degree −''i'' if :f(V_)\subseteq W_j for all ''j'' in ''I'', while :f(V_j)=0\, if is not in ''I''. Just as the set of linear maps from a vector space to itself forms an
associative algebra In mathematics, an associative algebra ''A'' is an algebraic structure with compatible operations of addition, multiplication (assumed to be associative), and a scalar multiplication by elements in some field ''K''. The addition and multiplic ...
(the algebra of endomorphisms of the vector space), the sets of homogeneous linear maps from a space to itself – either restricting degrees to ''I'' or allowing any degrees in the group ''A'' – form associative
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 ...
s over those index sets.


Operations on graded vector spaces

Some operations on vector spaces can be defined for graded vector spaces as well. Given two ''I''-graded vector spaces ''V'' and ''W'', their direct sum has underlying vector space ''V'' ⊕ ''W'' with gradation :(''V'' ⊕ ''W'')''i'' = ''Vi'' ⊕ ''Wi'' . If ''I'' is a
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'', ...
, then the tensor product of two ''I''-graded vector spaces ''V'' and ''W'' is another ''I''-graded vector space, V \otimes W, with gradation : (V \otimes W)_i = \bigoplus_ V_j \otimes W_k.


Hilbert–Poincaré series

Given a \N-graded vector space that is finite-dimensional for every n\in \N, its
Hilbert–Poincaré series In mathematics, and in particular in the field of algebra, a Hilbert–Poincaré series (also known under the name Hilbert series), named after David Hilbert and Henri Poincaré, is an adaptation of the notion of dimension to the context of grade ...
is the formal power series :\sum_\dim_K(V_n)\, t^n. From the formulas above, the Hilbert–Poincaré series of a direct sum and of a tensor product of graded vector spaces (finite dimensional in each degree) are respectively the sum and the product of the corresponding Hilbert–Poincaré series.


See also

* Graded (mathematics) *
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 ...
*
Comodule In mathematics, a comodule or corepresentation is a concept dual to a module. The definition of a comodule over a coalgebra is formed by dualizing the definition of a module over an associative algebra. Formal definition Let ''K'' be a field, an ...
*
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 ...
*
Littlewood–Richardson rule In mathematics, the Littlewood–Richardson rule is a combinatorial description of the coefficients that arise when decomposing a product of two Schur functions as a linear combination of other Schur functions. These coefficients are natural number ...


References

* Bourbaki, N. (1974) ''Algebra I'' (Chapters 1-3), , Chapter 2, Section 11; Chapter 3. {{DEFAULTSORT:Graded Vector Space Categories in category theory Vector spaces