Graded vector space - Citizendia

In mathematics, a graded vector space is a type of vector space that includes the extra structure of gradation, meaning that it can be composed into the direct sum of vector subspaces. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and In Mathematics, a vector space (or linear space) is a collection of objects (called vectors) that informally speaking may be scaled and added The symbol \oplus \! denotes direct sum it is also the astrological and astronomical symbol for Earth, and a symbol for the Exclusive disjunction

1 N-graded vector spaces
2 General I-graded vector spaces
3 Linear maps
4 See also

N-graded vector spaces

An $\mathbb{N}$ -graded vector space, often called simply a graded vector space without the prefix $\mathbb{N}$ , is a vector space V which decomposes into a direct sum of the form

$V = \bigoplus_{n \in \mathbb{N}} V_n$

where each $V n$ is a vector space. The symbol \oplus \! denotes direct sum it is also the astrological and astronomical symbol for Earth, and a symbol for the Exclusive disjunction 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 polynomials in one variable form a graded vector space, where the homogeneous elements of degree n are exactly the monomials of degree n. In Mathematics, a polynomial is an expression constructed from Variables (also known as indeterminates and Constants using the operations

General I-graded vector spaces

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 that can be written as a direct sum of subspaces indexed by elements i of set I:

$V = \bigoplus_{i \in I} V_i$ .

Therefore, an $\mathbb{N}$ -graded vector space, as defined above, is just an I-graded vector space where the set I is $\mathbb{N}$ (the set of natural numbers). In Mathematics, a natural number (also called counting number) can mean either an element of the set (the positive Integers or an

The case where I is the ring $\mathbb{Z}_2$ (the elements 0 and 1) is particularly important in physics. In Mathematics, a ring is an Algebraic structure which generalizes the algebraic properties of the Integers though the rational, real Physics (Greek Physis - φύσις in everyday terms is the Science of Matter and its motion. A $\mathbb{Z}_2$ -graded vector space also known as a supervector space. In Mathematics, a super vector space is another name for a Z 2- Graded vector space, that is a Vector space over a field

If I is a semigroup, then the tensor product of two I-graded vector spaces V and W is another I-graded vector space, $V \otimes W$

$(V \otimes W)_i = \bigoplus_{\{j,k|jk=i\}} V_j \otimes W_k$

Linear maps

When considering graded vector spaces, the nicest linear maps are those which respect the grading. In Mathematics, a semigroup is an Algebraic structure consisting of a nonempty set S together with an Associative Binary operation In Mathematics, the tensor product, denoted by \otimes may be applied in different contexts to vectors matrices, Tensors Vector In Mathematics, a linear map (also called a linear transformation, or linear operator) is a function between two Vector spaces that With this in mind, we define a linear map T between M-graded vector space V and N-graded vector space W to be such that for every m in M, there is some n in N with

$T(V_m)\subseteq W_n.$

Then the vector space L(V,W) of graded linear maps is itself an M×N-graded vector space, where M×N is the Cartesian product, since for each choice of homogeneous subspace in the domain, the map may choose a different range homogeneous subspace in the codomain. Cartesian square redirects here For Cartesian squares in Category theory, see Cartesian square (category theory. In Mathematics, the domain of a given function is the set of " Input " values for which the function is defined In Mathematics, the codomain, or target, of a function f: X → Y is the set

$L(V,W)=\bigoplus_{M\times N} L(V_m,W_n).$

For example, a linear map T between two Z₂-graded spaces can be decomposed into four parts: T₀₀ which carries even vectors to even vectors, T₁₀ which carries odd vectors to even vectors, T₀₁ which carries even vectors to odd vectors, and T₁₁ which carries odd vectors to odd vectors.

When the domain and codomain coincide, and if the grading set is a monoid which satisfies the cancellation law (for example, the natural numbers or any group), then one may define the graded map to be one which satisfies

$T_\alpha(V_\beta)\subseteq V_{\alpha\beta}$

This introduces a grading on this space of graded maps which is coarser than the grading available for whole space L(V,W) listed above, in the sense that the grading set of the latter is a subset of the former, and thus is compatible with it

$L_\beta=\bigoplus_{\alpha \in M}L(V_\alpha,V_{\beta\alpha})$

Because these homogeneous spaces include sums over all the subspaces of the domain, the elements may be considered as being defined over the whole domain, unlike the finer grading above. In Mathematics, the codomain, or target, of a function f: X → Y is the set In Abstract algebra, a branch of Mathematics, a monoid is an Algebraic structure with a single Associative Binary operation In Mathematics, the notion of cancellative is a generalization of the notion of Invertible. In Mathematics, a natural number (also called counting number) can mean either an element of the set (the positive Integers or an In Mathematics, a group is a set of elements together with an operation that combines any two of its elements to form a third element Thus when the domain and codomain coincide, composition of maps is always defined. By construction, these graded maps satisfy

$L_\alpha L_\beta\subseteq L_{\alpha\beta},$

so that just as the set of linear maps from a vector space to itself forms an associative algebra (the algebra of endomorphisms of the vector space), the graded linear maps from a space to itself forms an associative graded algebra. In Mathematics, an associative algebra is a Vector space (or more generally a module) which also allows the multiplication of vectors in a distributive In Mathematics, an endomorphism is a Morphism (or Homomorphism) from a mathematical object to itself In Mathematics, in particular Abstract algebra, a graded algebra is an Algebra over a field (or Commutative ring) with an extra piece of structure This is the benefit of restricting to only the graded maps.

Contents

N-graded vector spaces

General I-graded vector spaces

Linear maps

See also