Graded vector space

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In mathematics, a graded vector space is a type of vector space that includes the extra structure of gradation, which is a decomposition of the vector space into a direct sum of vector subspaces.

N-graded vector spaces[edit]

Let \mathbb{N} be the set of non-negative integers. 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. 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.

General I-graded vector spaces[edit]

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).

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.

Linear maps[edit]

For general index sets I, a linear map between two I-graded vector spaces f:VW is called a graded linear map if it preserves the grading of homogeneous elements:

f(V_i)\subseteq W_i for all i in I.

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 a commutative group A which 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). In particular 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 ji 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[edit]

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 VW with gradation

(VW)i = ViWi .

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_{\{j,k|j+k=i\}} V_j \otimes W_k.

See also[edit]

References[edit]