Definitions

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.

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

$V = bigoplus_\left\{n in mathbb\left\{N\right\}\right\} V_n$
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 polynomials in one variable form a graded vector space, where the homogeneous elements of degree n are exactly the linear combinations of monomials of degree n.

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:

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

Therefore, an $mathbb\left\{N\right\}$-graded vector space, as defined above, is just an I-graded vector space where the set I is $mathbb\left\{N\right\}$ (the set of natural numbers).

The case where I is the ring $mathbb\left\{Z\right\}_2$ (the elements 0 and 1) is particularly important in physics. A $mathbb\left\{Z\right\}_2$-graded vector space also known as a supervector space.

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

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

## Linear maps

When considering graded vector spaces, the nicest linear maps are those which respect the grading. With this in mind, we define a linear map T from the M-graded vector space V to the N-graded vector space W to be such that for every m in M, there is some n in N with

$T\left(V_m\right)subseteq W_n.$

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.

$L\left(V,W\right)=bigoplus_\left\{Mtimes N\right\} L\left(V_m,W_n\right).$

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

$T_alpha\left(V_beta\right)subseteq V_\left\{alphabeta\right\}$

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

$L_beta=bigoplus_\left\{alpha in M\right\}L\left(V_alpha,V_\left\{betaalpha\right\}\right)$

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. Thus when the domain and codomain coincide, composition of maps is always defined. By construction, these graded maps satisfy

$L_alpha L_betasubseteq L_\left\{alphabeta\right\},$

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. This is the benefit of restricting to only the graded maps.

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