Graded vector space

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:

${\displaystyle V=\underset{i\in I}{⨁}{V}_i.}$

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

The case where I is the ring ${\displaystyle \mathbb{Z}/2\mathbb{Z}}$ (the elements 0 and 1) is particularly important in physics. A ${\displaystyle (\mathbb{Z}/2\mathbb{Z})}$-graded vector space is also known as a supervector space.

Homomorphisms

For general index sets I, a linear map between two I-graded vector spaces f : V → W 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:

${\displaystyle f(V_i)\subseteq W_i}$ for all i in I.

For a fixed field and a fixed index set, the graded vector spaces form a category whose morphisms are the graded linear maps.

When I is a commutative monoid (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

${\displaystyle f(V_j)\subseteq W_{i+j}}$ for all j in I,

where "+" denotes the monoid operation. If moreover I satisfies the cancellation property so that it can be embedded into an abelian group 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

${\displaystyle f(V_{i+j})\subseteq W_j}$ for all j in I, while

${\displaystyle f(V_j)=0}$ if j − i is not in I.

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 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 algebras 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 = V_i ⊕ W_i .

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

${\displaystyle (V\otimes W{)}_i=\underset{\left\{\left(j,k\right):j+k=i\right\}}{⨁}{V}_j\otimes W_k.}$

Hilbert–Poincaré series

Given a ${\displaystyle \mathbb{N}}$-graded vector space that is finite-dimensional for every ${\displaystyle n\in \mathbb{N},}$ its Hilbert–Poincaré series is the formal power series

${\displaystyle \sum _{n\in \mathbb{N}}{\dim }_K(V_n)t^n.}$

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.