Graded vector space

Last updated October 01, 2024

In mathematics, a graded vector space is a vector space that has the extra structure of a grading or gradation, which is a decomposition of the vector space into a direct sum of vector subspaces, generally indexed by the integers.

Integer gradation

Let $\mathbb {N}$ be the set of non-negative integers. An ${\textstyle \mathbb {N} }$ -graded vector space, often called simply a graded vector space without the prefix $\mathbb {N}$ , is a vector space $V$ together with a decomposition into a direct sum of the form

V=\bigoplus _{n\in \mathbb {N} }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 or several variables forms a graded vector space, where the homogeneous elements of degree n are exactly the linear combinations of monomials of degree 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 _{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).

The case where I is the ring $\mathbb {Z} /2\mathbb {Z}$ (the elements 0 and 1) is particularly important in physics. A $(\mathbb {Z} /2\mathbb {Z} )$ -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:

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

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

f(V_{i+j})\subseteq W_{j}

for all j in I, while

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, $V\otimes W$ , with gradation

(V\otimes W)_{i}=\bigoplus _{\left\{\left(j,k\right)\,:\;j+k=i\right\}}V_{j}\otimes W_{k}.

Hilbert–Poincaré series

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

\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.

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References

Bourbaki, N. (1974) Algebra I (Chapters 1-3), ISBN 978-3-540-64243-5, Chapter 2, Section 11; Chapter 3.

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