graded vector space

In mathematics, a graded vector space is a vector space with an extra piece of structure, known as a grading.

Graded vector spaces

A graded vector space is a vector space V which can be written as a direct sum of the form

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

for each natural number n. The elements of

V_i

are known as 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 polynomials of degree n.

I-graded vector spaces

I-graded vector spaces generalize graded vector spaces. Let I be a set. An I-graded vector space V is a vector space that can be written as a direct sum of subspaces indexed by I:

V = \bigoplus_{i \in I} V_i

A graded vector space, as defined above, is just an N-graded vector space, where N is the set of natural numbers. The case when I=Z₂ is particularly important in physics. A Z₂-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

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

Graded Vector Space

Graded vector spaces

I-graded vector spaces

See also