Linear algebra MOC

Graded vector space

Given a set 𝑆, a vector space 𝑉 over 𝕂 is said to be 𝑆-graded iff it is (canonically) the internal direct sum linalg

𝑉=∐𝛼∈𝑆𝑉𝛼

for so-called homogenous subspaces 𝑉𝛼 of degrees 𝛼 ∈𝑆, the elements whereof are called homogenous elements of degree 𝛼 ∈𝑆.¹ For 𝑣 ∈𝑉𝛼, we write

deg⁡𝑣=𝛼

A graded vector space is thus a Graded module over a field (with the trivial gradation), where 𝑆 may take arbitrary monoidal structure.

One often expresses the dimensions of homogenous subspaces as a formal power series, called the Graded dimension.

Category of graded vector spaces

Many of our typical vector space constructions carry over nicely, although some require monoid structure on 𝑆. These motivate the categories Strict category of graded vector spaces and Closed category of graded vector spaces.

• Homomorphism of graded vector spaces
• Graded vector subspace, Quotient graded vector space
• Direct sum of graded vector spaces
• Tensor product of graded vector spaces

See also

• Degree operator
• Graded structure

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Footnotes

1. 1988. Vertex operator algebras and the Monster, p. 8 ↩