Linear algebra MOC

Graded vector space

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

for so-called homogenous subspaces of degrees , the elements whereof are called homogenous elements of degree .¹ For , we write

A graded vector space is thus a Graded module over a field (with the trivial gradation), where 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 . 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 ↩