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

<math>V = \bigoplus_{n \in \mathbb{N}} V_n<math>

for each natural number n. The elements of <math>V_i<math> 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:

<math>V = \bigoplus_{i \in I} V_i<math>.

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, <math>V \otimes W<math>

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

See also

• graded algebra
• comodule